{"id":4668,"date":"2022-03-16T02:42:43","date_gmt":"2022-03-15T21:12:43","guid":{"rendered":"https:\/\/envisionmathanswerkey.com\/?p=4668"},"modified":"2022-03-16T16:03:48","modified_gmt":"2022-03-16T10:33:48","slug":"envision-math-common-core-grade-8-answer-key-topic-1","status":"publish","type":"post","link":"https:\/\/envisionmathanswerkey.com\/envision-math-common-core-grade-8-answer-key-topic-1\/","title":{"rendered":"enVision Math Common Core Grade 8 Answer Key Topic 1 Real Numbers"},"content":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 8 Answer Key<\/a>\u00a0Topic 1 Real Numbers<\/strong>\u00a0regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 8th Grade Answers Key Topic 1 Real Numbers<\/h2>\n

enVision STEM Project<\/strong><\/p>\n

Did You Know?<\/strong>
\nNatural resources are materials that occur in nature, such as water, fossil fuels, wood, and minerals. Natural resources not only meet basic human needs, but also support industry and economy.
\n\"Envision
\nMinerals are used in the manufacturing of all types of common objects, including cell phones, computers, light bulbs, and medicines.
\n\"Envision
\nWater, oil, and forests are some of the natural resources that are in danger of someday being depleted.
\n70% of available fresh water is used in agriculture…
\n\"Envision
\n…and 10% for human consumption.<\/p>\n

Each person in the United States needs over 48,000 pounds of minerals each year.
\n\"Envision<\/p>\n

About 18 million acres of forest are lost to deforestation each year.
\n\"Envision<\/p>\n

Solar power, wind power and other renewable energy sources are helping to lessen the dependency on oil and fossil fuels.
\n\"Envision<\/p>\n

Fossil fuels are expected to supply almost 80% of world energy use through 2040.
\n\"Envision<\/p>\n

Your Task: Going, Going, Gone?<\/strong><\/p>\n

Natural resource depletion is an important issue facing the world. Suppose a natural resource is being depleted at the rate of 1.333% per year. If there were 300 million tons of this resource in 2005, and there are no new discoveries, how much will be left in the year 2045? You and your classmates will explore the depletion of this resource over time.
\n\"Envision
\nAnswer:
\nIt is given that
\nA natural resource is being depleted at the rate of 1.333% per year and there were 300 million tons of this resource in 2005, and there are no new discoveries
\nSo,
\nThe number of resources left in 2045 = The number of resources present in 2005 – The decrease of the number of resources from 2005 to 2045
\nSo,
\nThe number of resources left in 2045 = 300 million – 1.33% of 300 million \u00d7 (2045 – 2005)
\nThe number of resources left in 2045 = 300 million – 159.6 million
\nThe number of resources left in 2045 = 140.4 million
\nHence, from the above,
\nWe can conclude that the number of resources left in 2045 is: 140.4 million<\/p>\n

Topic 1 GET READY!<\/h3>\n

Review What You Know!<\/strong><\/p>\n

Vocabulary<\/strong><\/p>\n

Choose the best term from the box. Write it on the blank.<\/strong><\/p>\n

Question 1.
\nA(n) ____ is a decimal that ends in repeating zeros.
\nAnswer:
\nA “Terminating decimal” is a decimal that ends in repeating zeros
\nHence, from the above,
\nWe can conclude that the best term from the box for this definition is “Terminating decimal”<\/p>\n

Question 2.
\nA(n) ____ is a decimal in which a digit or digits repeat
\nAnswer:
\nA “Repeating decimal” is a decimal in which a digit or digits repeat
\nHence, from the above,
\nWe can conclude that the best term from the box for this definition is “Repeating decimal”<\/p>\n

Question 3.
\nA(n) ____ is either a counting number, the opposite of a counting number, or zero
\nAnswer:
\nAn “Integer” is either a counting number, the opposite of a counting number, or zero
\nHence, from the above,
\nWe can conclude that the best term from the box for this definition is “Integer”<\/p>\n

Question 4.
\nA(n) ___ is a number that can be used to describe a part of a whole, a part of a set, a location on a number line, or a division of whole numbers.
\nAnswer:
\nA “Fraction” is a number that can be used to describe a part of a whole, a part of a set, a location on a number line, or a division of whole numbers.
\nHence, from the above,
\nWe can conclude that the best term from the box for this definition is “Fraction”<\/p>\n

Terminating and Repeating Decimals<\/strong><\/p>\n

Determine whether each decimal is terminating or repeating.<\/p>\n

Question 5.
\n5.692
\nAnswer:
\nThe given decimal is: 5.692
\nWe know that,
\nA\u00a0terminating decimal\u00a0is usually defined as a\u00a0decimal\u00a0number that contains a finite number of digits after the\u00a0decimal\u00a0point
\nHence, from the above,
\nWe can conclude that the given decimal is a terminating decimal<\/p>\n

Question 6.
\n-0.222222…
\nAnswer:
\nThe given decimal is -0.222222….
\nWe know that,
\nA repeating decimal or recurring decimal is the decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero
\nHence, from the above,
\nWe can conclude that the given decimal is a repeating decimal<\/p>\n

Question 7.
\n7.0001
\nAnswer:
\nThe given decimal is: 7.0001
\nWe know that,
\nA\u00a0terminating decimal\u00a0is usually defined as a\u00a0decimal\u00a0number that contains a finite number of digits after the\u00a0decimal\u00a0point
\nHence, from the above,
\nWe can conclude that the given decimal is a terminating decimal<\/p>\n

Question 8.
\n7.2\\(\\overline{8}\\)
\nAnswer:
\nThe given decimal is: 7.2\\(\\overline{8}\\)
\nWe know that,
\nA repeating decimal or recurring decimal is the decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero
\nHence, from the above,
\nWe can conclude that the given decimal is a repeating decimal<\/p>\n

Question 9.
\n1.\\(\\overline{178}\\)
\nAnswer:
\nThe given decimal is: 1.\\(\\overline{178}\\)
\nWe know that,
\nA repeating decimal or recurring decimal is the decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero
\nHence, from the above,
\nWe can conclude that the given decimal is a repeating decimal<\/p>\n

Question 10.
\n-4.03479
\nAnswer:
\nThe given decimal is: -4.03479
\nWe know that,
\nA\u00a0terminating decimal\u00a0is usually defined as a\u00a0decimal\u00a0number that contains a finite number of digits after the\u00a0decimal\u00a0point
\nHence, from the above,
\nWe can conclude that the given decimal is a terminating decimal<\/p>\n

Multiplying Integers<\/strong><\/p>\n

Find each product.<\/p>\n

Question 11.
\n2.2
\nAnswer:
\nThe given expression is: 2 \u00d7 2
\nHence,
\n2 \u00d7 2 = 4<\/p>\n

Question 12.
\n-5. (-5)
\nAnswer:
\nThe given expression is: -5 \u00d7 -5
\nWe know that,
\n– \u00d7 – = +
\nHence,
\n-5 \u00d7 -5 = 25<\/p>\n

Question 13.
\n7.7
\nAnswer:
\nThe given expression is: 7 \u00d7 7
\nHence,
\n7 \u00d7 7 = 49<\/p>\n

Question 14.
\n-6 \u2219 (-6) \u2219 (-6)
\nAnswer:
\nThe given expression is: -6 \u00d7 -6 \u00d7 -6
\nWe know that,
\n– \u00d7 – \u00d7 –
\n= + \u00d7 –
\n= –
\nHence,
\n-6 \u00d7 -6 \u00d7 -6
\n= 36 \u00d7 -6
\n= -216<\/p>\n

Question 15.
\n10 \u2219 10 \u2219 10
\nAnswer:
\nThe given expression is: 10 \u00d7 10 \u00d7 10
\nHence,
\n10 \u00d7 10 \u00d7 10 = 1,000<\/p>\n

Question 16.
\n-9 \u2219 (-9) \u2219 (-9)
\nAnswer:
\nThe given expression is: -9 \u00d7 -9 \u00d7 -9
\nWe know that,
\n– \u00d7 – \u00d7 –
\n= + \u00d7 –
\n= –
\nHence,
\n-9 \u00d7 -9 \u00d7 -9
\n= 81 \u00d7 -9
\n= -729<\/p>\n

Simplifying Expressions<\/strong>
\nSimplify each expression.<\/strong><\/p>\n

Question 17.
\n(4 \u2219 10) + (5 \u2219 100)
\nAnswer:
\nThe given expression is:
\n(4 \u00d7 10) + (5 \u00d7 100)
\nSo,
\n(4 \u00d7 10) + (5 \u00d7 100)
\n= 40 + 500
\n= 540
\nHence, from the above,
\nWe can conclude that the value of the expression is: 540<\/p>\n

Question 18.
\n(2100) + (7.10)
\nAnswer:
\nThe given expression is:
\n2100 + (7 \u00d7 10)
\nSo,
\n2100 + (7 \u00d7 10)
\n= 2100 + 70
\n= 2170
\nHence, from the above,
\nWe can conclude that the value of the given expression is: 2170<\/p>\n

Question 19.
\n(6 \u00b7 100) \u2013 (1 \u00b7 10)
\nAnswer:
\nThe given expression is:
\n(6 \u00d7 100) – (1 \u00d7 10)
\nSo,
\n(6 \u00d7 100) – (1 \u00d7 10)
\n= 600 – 10
\n= 590
\nHence, from the above,
\nWe can conclude that the value of the given expression is: 590<\/p>\n

Question 20.
\n(9 \u2219 1,000) + (4 \u2219 10)
\nAnswer:
\nThe given expression is:
\n(9 \u00d7 1,000) + (4 \u00d7 10)
\nSo,
\n(9 \u00d7 1,000) + (4 \u00d7 10)
\n= 9,000 + 40
\n= 9,040
\nHence, from the above,
\nWe can conclude that the value of the given expression is: 9,040<\/p>\n

Question 21.
\n(3 \u00b7 1,000) – (2 \u2219 100)
\nAnswer:
\nThe given expression is:
\n(3 \u00d7 1,000) – (2 \u00d7 100)
\nSo,
\n(3 \u00d7 1,000) – (2 \u00d7 100)
\n= 3,000 – 200
\n= 2,800
\nHence, from the above,
\nWe can conclude that the value of the given expression is: 2,800<\/p>\n

Question 22.
\n(2 \u2219 10) + (7 \u00b7 100)
\nAnswer:
\nThe given expression is:
\n(2 \u00d7 10) + (7 \u00d7 100)
\nSo,
\n(2 \u00d7 10) + (7 \u00d7 100)
\n= 20 + 700
\n= 720
\nHence, from the above,
\nwe can conclude that the value of the given expression is: 720<\/p>\n

Language Development<\/p>\n

Fill in the word map with new terms, definitions, and supporting examples or illustrations.<\/p>\n

\"Envision
\n\"Envision
\n\"Envision<\/p>\n

Topic 1 PICK A PROJECT<\/h3>\n

PROJECT 1A<\/strong>
\nWho is your favorite poet, and why?
\nPROJECT: WRITE A POEM
\n\"Envision<\/p>\n

PROJECT 1B<\/strong>
\nIf you moved to a tiny house, what would you bring with you?
\nPROJECT: DESIGN A TINY HOUSE
\n\"Envision<\/p>\n

PROJECT 1C<\/strong>
\nIf you could travel anywhere in space, where would you go?
\nPROJECT: PLAN A TOUR OF THE MILKY WAY
\n\"Envision<\/p>\n

PROJECT 1D<\/strong>
\nWhy do you think people tell stories around a campfire?
\nPROJECT: TELL A FOLK STORY
\n\"Envision<\/p>\n

Lesson 1.1 Rational Numbers as Decimals<\/p>\n

Solve & Discuss It!<\/strong><\/p>\n

Jaylon has a wrench labeled 0.1875 inches and bolts labeled in fractions of an inch. Which size bolt will fit best with the wrench? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nJaylon has a wrench labeled 0.1875 inches and bolts labeled in fractions of an inch.
\nNow,
\nWe know that,
\nThe bolt will be fit in a wrench only when
\nThe size of the bolt (inches) = The size of the wrench (inches)
\nNow,
\nThe representation of the bolts in the decimal numbers is:
\n\\(\\frac{3}{8}\\) = 0.375 inches
\n\\(\\frac{1}{8}\\) = 0.046 inches
\n\\(\\frac{3}{16}\\) = 0.1875 inches
\n\\(\\frac{1}{4}\\) = 0.25 inches
\nHence, from the above,
\nWe can conclude that the bolt which has the size \\(\\frac{3}{16}\\) inches will fit best with the wrench<\/p>\n

Reasoning<\/strong>
\nHow can you write these numbers in the same form?
\nAnswer:
\nThe representation of the sizes of bolts in the decimal form is by using the properties of place values
\nWe know that,
\nA terminating decimal can\u00a0be\u00a0<\/span>written as a fraction\u00a0by using properties of place value.
\nExample:
\n3.75 = three and seventy-five hundredths or \\(\\frac{375}{100}\\), which is equal to the improper <\/span>fraction<\/p>\n

Focus on math practices<\/strong>
\nReasoning Why is it useful to write a rational number as a fraction or as a decimal?
\nAnswer:
\nRational numbers\u00a0are whole\u00a0numbers,\u00a0fractions, and\u00a0decimals\u00a0– the\u00a0numbers\u00a0we use in our daily lives. They can be written as a ratio of two integers. … The definition says that a\u00a0number is rational if you can write it in the form \\(\\frac{a}{b}\\) where a and b are integers, and b is not zero.<\/p>\n

? Essential Question<\/strong>
\nHow can you write repeating decimals as fractions?
\nAnswer:
\nBecause repeating decimals are rational numbers, you can write them in fraction form.
\nSTEP 1
\nAssign a variable to represent the repeating decimal.
\nSTEP 2
\nWrite an equation: variable = decimal.
\nSTEP 3
\nMultiply each side of the equation by 10d<\/sup>, where d is the number of repeating digits in the repeating decimal.
\nSTEP 4
\nSubtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
\nSTEP 5
\nSolve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.<\/p>\n

Try It!<\/strong><\/p>\n

In another baseball division, one team had a winning percentage of 0.444…. What fraction of their games did this team win?
\nThe team won their games.
\n\"Envision
\nAnswer:
\nIt is given that
\nIn another baseball division, one team had a winning percentage of 0.444…
\nSince only 1 number is repeated,
\nThe repeating decimal can be written as 0.\\(\\overline{4}\\)
\nNow,
\nAccording to the steps of converting a repeating decimal to a fraction (Essential Question),
\nStep 1:
\nLet x = 0.\\(\\overline{4}\\)
\nStep 2:
\nMultiply with 10 on both sides since only 1 number is repeating
\nSo,
\n10x = 10 (0.\\(\\overline{4}\\))
\n10x = 4.\\(\\overline{4}\\)
\nStep 3:
\n10x – x = 4.\\(\\overline{4}\\) – 0.\\(\\overline{4}\\)
\n9x = 4
\nx = \\(\\frac{4}{9}\\)
\nHence, from the above,
\nWe can conclude that the team won \\(\\frac{4}{9}\\) of their games<\/p>\n

Convince Me!<\/strong>
\nHow do you know what power of ten to multiply by in the second step at the right?
\nAnswer:
\nLet x be the repeating portion.
\nMultiply this equation by a power of 10 to move the repeating digits immediately to the left of the decimal\u00a0point (in other words, to eliminate any zeros preceding the\u00a0repeating digits).<\/p>\n

Try It!<\/strong>
\nWrite the repeating decimal 0.63333… as a fraction.
\nAnswer:
\nThe given repeating decimal is: 0.6333333
\nNow,
\nAccording to the steps of converting a repeating decimal to a fraction (Essential Question),
\nStep 1:
\nLet x = 0.6\\(\\overline{3}\\)
\nStep 2:
\nMultiply with 10 on both sides since only 1 number is repeating
\nSo,
\n10x = 10 (0.6\\(\\overline{3}\\))
\n10x = 6.\\(\\overline{3}\\)
\nStep 3:
\n10x – x = 6.\\(\\overline{3}\\) – 0.6\\(\\overline{3}\\)
\n9x = 6.33 – 0.63
\n9x = 5.7
\nDivide by 9 into both sides
\nSo,
\n\\(\\frac{9}{9}\\)x = \\(\\frac{5.7}{9}\\)
\nx = \\(\\frac{57}{90}\\)
\nHence, from the above,
\nWe can conclude that the value of the given repeating decimal in the fraction form is: \\(\\frac{57}{90}\\)<\/p>\n

Try It!<\/strong>
\nWrite the repeating decimal 4.1363636… as a fraction.
\nAnswer:
\nThe given repeating decimal is 4.1363636…..
\nNow,
\nAccording to the steps of converting a repeating decimal to a fraction (Essential Question),
\nStep 1:
\nLet x = 4.1\\(\\overline{36}\\)
\nStep 2:
\nMultiply with 100 on both sides since 2 numbers are repeating
\nSo,
\n100x = 100 (4.1\\(\\overline{36}\\))
\n100x = 41.\\(\\overline{36}\\)
\nStep 3:
\n100x – x = 41.\\(\\overline{36}\\) – 4.1\\(\\overline{36}\\)
\n99x = 413.636 – 4.136
\n99x = 409.5
\nDivide by 99 into both sides
\nSo,
\n\\(\\frac{99}{99}\\)x = \\(\\frac{409.5}{99}\\)
\nx = \\(\\frac{4095}{990}\\)
\nHence, from the above,
\nWe can conclude that the value of the given repeating decimal in the fraction form is: \\(\\frac{4095}{990}\\)<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

Because repeating decimals are rational numbers, you can write them in fraction form.
\nSTEP 1<\/strong>
\nAssign a variable to represent the repeating decimal.
\nSTEP 2<\/strong>
\nWrite an equation: variable = decimal.
\nSTEP 3<\/strong>
\nMultiply each side of the equation by 10d<\/sup>, where d is the number of repeating digits in the repeating decimal.
\nSTEP 4<\/strong>
\nSubtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
\nSTEP 5<\/strong>
\nSolve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong>
\nHow can you write repeating decimals as fractions?
\nAnswer:
\nBecause repeating decimals are rational numbers, you can write them in fraction form.
\nSTEP 1
\nAssign a variable to represent the repeating decimal.
\nSTEP 2
\nWrite an equation: variable = decimal.
\nSTEP 3
\nMultiply each side of the equation by 10d<\/sup>, where d is the number of repeating digits in the repeating decimal.
\nSTEP 4
\nSubtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
\nSTEP 5
\nSolve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.<\/p>\n

Question 2.
\nUse Structure<\/strong> Why do you multiply by a power of 10 when writing a repeating decimal as a rational number?
\nAnswer:
\nThe idea is to\u00a0multiply\u00a0by some\u00a0number\u00a0(10, 100, 1000, etc.) so that when\u00a0we\u00a0subtract the original\u00a0number\u00a0from the multiple, the\u00a0repeating\u00a0part cancels out.<\/p>\n

Question 3.
\nBe Precise<\/strong> How do you decide by which power of 10 to multiply an equation when writing a decimal with repeating digits as a fraction?
\nAnswer:
\nThe idea is to\u00a0multiply\u00a0by some\u00a0number\u00a0(10, 100, 1000, etc.) so that when\u00a0we\u00a0subtract the original\u00a0number\u00a0from the multiple, the\u00a0repeating\u00a0part cancels out.<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nA survey reported that 63.63% of moviegoers prefer action films. This percent represents a repeating decimal. Write it as a fraction.
\nAnswer:
\nIt is given that
\nA survey reported that 63.63% of moviegoers prefer action films. This percent represents a repeating decimal
\nSo,
\nThe given repeating decimal is 63.6363…..%
\nNow,
\nAccording to the steps of converting a repeating decimal to a fraction (Essential Question),
\nStep 1:
\nLet x = 63.\\(\\overline{63}\\)
\nStep 2:
\nMultiply with 100 on both sides since 2 numbers are repeating
\nSo,
\n100x = 100 (63.\\(\\overline{63}\\))
\n100x = 6363.\\(\\overline{63}\\)
\nStep 3:
\n100x – x = 6363.\\(\\overline{63}\\) – 63.\\(\\overline{63}\\)
\n99x = 6,300
\nDivide by 99 into both sides
\nSo,
\n\\(\\frac{99}{99}\\)x = \\(\\frac{6,300}{99}\\)
\nx = \\(\\frac{6,300}{99}\\)
\nHence, from the above,
\nWe can conclude that the value of the given repeating decimal in the fraction form is: \\(\\frac{6,300}{99}\\)<\/p>\n

Question 5.
\nA student estimates the weight of astronauts on the Moon by multiplying their weight by the decimal 0.16666… What fraction can be used for the same estimation?
\n\"Envision
\nAnswer:
\nIt is given that
\nA student estimates the weight of astronauts on the Moon by multiplying their weight by the decimal 0.16666…
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that the fraction that can be used for the same estimation is: \\(\\frac{1}{6}\\)<\/p>\n

Question 6.
\nWrite 2.3181818… as a mixed number.
\nAnswer:
\nThe given repeating decimal is 2.3181818…
\nNow,
\nAccording to the steps of converting a repeating decimal to a fraction (Essential Question),
\nStep 1:
\nLet x = 2.3\\(\\overline{18}\\)
\nStep 2:
\nMultiply with 100 on both sides since 2 numbers are repeating
\nSo,
\n100x = 100 (2.3\\(\\overline{18}\\))
\n100x = 23.\\(\\overline{18}\\)
\nStep 3:
\n100x – x = 23.\\(\\overline{18}\\) – 2.3\\(\\overline{18}\\)
\n99x = 231.818 – 2.318
\n99x = 229.5
\nDivide by 99 into both sides
\nSo,
\n\\(\\frac{99}{99}\\)x = \\(\\frac{229.5}{99}\\)
\nx = \\(\\frac{2295}{990}\\)
\nx = \\(\\frac{51}{22}\\)
\nSo,
\nThe representation of the above fraction in the mixed form is: 2\\(\\frac{7}{22}\\)
\nHence, from the above,
\nWe can conclude that the value of the given repeating decimal in the mixed fraction form is: 2\\(\\frac{7}{22}\\)<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice In 7 and 8, write the decimal as a fraction or mixed number.<\/p>\n

Question 7.
\nWrite the number 0.21212121… as a fraction.
\nLet x = \"Envision
\n100x = \"Envision
\n100x – x = \"Envision – \"Envision
\n99x = \"Envision
\nx = \"Envision
\nSo 0.2121… is equal to
\nAnswer:
\nThe given repeating decimal is: 0.212121….
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that the value of the repeating decimal in the fraction form is: \\(\\frac{21}{99}\\)<\/p>\n

Question 8.
\nWrite 3.7 as a mixed number.
\nLet x = \"Envision
\n10x = \"Envision
\n9x = \"Envision
\nx = \"Envision
\nSo 3.\\(\\overline{7}\\) is equal to \"Envision
\nAnswer:
\nThe given repeating decimal is: 3.\\(\\overline{7}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that the representation of the given repeating decimal in the mixed fraction form is: 3\\(\\frac{7}{9}\\)<\/p>\n

Question 9.
\nWrite the number shown on the scale as a fraction.
\n\"Envision
\nAnswer
\nThe number that is shown on the scale is: 0.233333
\nNow,
\nFrom the: above number,
\nWe can observe that the number is a repeating decimal
\nNow,
\n\"\"
\nSo,
\nThe simplified form of \\(\\frac{2.1}{9}\\) is: \\(\\frac{7}{30}\\)
\nHence, from the above,
\nWe can conclude that the representation of the number that is shown on the scale as a fraction is: \\(\\frac{7}{30}\\)<\/p>\n

Question 10.
\nTomas asked 15 students whether summer break should be longer. He used his calculator to divide the number of students who said yes by the total number of students. His calculator showed the result as 0.9333…
\na. Write this number as a fraction.
\nAnswer:
\nThe given repeating decimal is 0.93333…
\nNow,
\n\"\"
\nSo,
\nThe simplified form of \\(\\frac{8.4}{9}\\) is: \\(\\frac{14}{15}\\)
\nHence, from the above,
\nWe can conclude that the representation of the repeating number in the form of the fraction is: \\(\\frac{14}{15}\\)<\/p>\n

b. How many students said that summer break should be longer?
\nAnswer:
\nIt is given that
\nTomas asked 15 students whether summer break should be longer. He used his calculator to divide the number of students who said yes by the total number of students.
\nSo,
\nThe number of students that said summer break should be longer = \\(\\frac{The number of students that said yes that summer break is longer}{The total number of students}
\nNow,
\nFrom part (a),
\nThe fraction form of his calculated result from part (a) is: 14 \/ 15
\nHence, from the above,
\nWe can conclude that the number of students that said the summer break should be longer is: 14 students<\/p>\n

Question 11.
\nWrite 0.[latex]\\overline{87}\\) as a fraction.
\nAnswer:
\nThe given repeating decimal is 0.\\(\\overline{87}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that the representation of the repeating decimal in the fraction form in the simplest form is: \\(\\frac{29}{33}\\)<\/p>\n

Question 12.
\nWrite 0.\\(\\overline{8}\\) as a fraction.
\nAnswer:
\nThe given repeating decimal is 0.\\(\\overline{8}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that the representation of the given repeating decimal in the fraction form is: \\(\\frac{8}{9}\\)<\/p>\n

Question 13.
\nWrite 1.\\(\\overline{48}\\) as a mixed number.
\nAnswer:
\nThe given repeating decimal number is 1.\\(\\overline{48}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that the representation of the given repeating decimal in the mixed fraction form is: 1\\(\\frac{16}{33}\\)<\/p>\n

Question 14.
\nWrite 0.\\(\\overline{6}\\) as a fraction.
\nAnswer:
\nThe given repeating decimal is 0.\\(\\overline{6}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that the representation of the given repeating decimal in the fraction form is: \\(\\frac{2}{3}\\)<\/p>\n

Question 15.
\nA manufacturer determines that the cost of making a computer component is $2.161616. Write the cost as a fraction and as a mixed number.
\n\"Envision
\nAnswer:
\nIt is given that
\nA manufacturer determines that the cost of making a computer component is $2.161616
\nSo,
\nThe given repeating decimal is 2.161616…
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe cost of a computer component in a fraction form is: \\(\\frac{214}{99}\\)
\nThe cost of a computer component in a mixed fraction form is: 2\\(\\frac{16}{99}\\)<\/p>\n

Question 16.
\nReasoning<\/strong> When writing a repeating decimal as a fraction, does the number of repeating digits you use matter? Explain.
\nAnswer:
\nNo. Even if the\u00a0number\u00a0of different\u00a0digits\u00a0in the cycle is 1 or 1 million, the method of finding the\u00a0fraction\u00a0is the same<\/p>\n

Question 17.
\nHigher Order Thinking<\/strong> When writing a repeating decimal as a fraction, why does the fraction always have only 9s or 9s and 0s as digits in the denominator?
\nAnswer:
\nWhen writing a repeating decimal as a fraction, the fraction always has only 9s or 9s and 0s as digits in the denominator because we are talking here about a geometric series and they are decimals, so the right side i.e., after the decimal point, the digits are in tenths, hundredths and so on<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 18.
\nWhich decimal is equivalent to \\(\\frac{188}{11}\\)?
\nA. 17.\\(\\overline{09}\\)
\nB. 17.0\\(\\overline{09}\\)
\nC. 17.\\(\\overline{1709}\\)
\nD. 17.\\(\\overline{1709}\\)0
\nAnswer:
\nThe given fraction is: \\(\\frac{188}{11}\\)
\nSo,
\nThe representation of the given fraction in the decimal form is:
\n\"\"
\nwe know that,
\n\\(\\frac{1}{11}\\) = 0.090909…..
\nSo,
\n\\(\\frac{188}{11}\\) = 17.090909….
\n= 17.\\(\\overline{09}\\)
\nHence, from the above,
\nWe can conclude that option A matches the representation of the repeating decimal for the given fraction<\/p>\n

Question 19.
\nChoose the repeating decimal that is equal to the fraction on the left.
\n\"Envision
\nAnswer:
\nFollow the process that is mentioned below to solve the given repeating decimals in the fraction form
\nNow,
\nSTEP 1
\nAssign a variable to represent the repeating decimal.
\nSTEP 2
\nWrite an equation: variable = decimal.
\nSTEP 3
\nMultiply each side of the equation by 10d<\/sup>, where d is the number of repeating digits in the repeating decimal.
\nSTEP 4
\nSubtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
\nSTEP 5
\nSolve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.
\nHence,
\n\"\"<\/p>\n

Lesson 1.2 Understand Irrational Numbers<\/h3>\n

Explain It!<\/strong>
\nSofia wrote a decimal as a fraction. Her classmate Nora says that her method and answer are not correct. Sofia disagrees and says that this is the method she learned.
\n\"Envision<\/p>\n

A. Construct Arguments<\/strong> Is Nora or Sofia correct? Explain your reasoning.
\nAnswer:
\nThe given number is 0.121121112111112…..
\nNow,
\nFrom the given number,
\nWe can observe that the given decimal is not a repeating decimal because there are other numbers other than the repeating numbers in the given decimal or a terminating decimal because the decimal is not finite
\nSo,
\nSince the given decimal is not a repeating decimal,
\nThe method that we used to convert the repeating decimal into a fraction is not applicable
\nHence, from the above,
\nWe can conclude that Nora is correct<\/p>\n

B. Use Structure<\/strong> What is a nonterminating decimal number that can not be written as a fraction.
\nAnswer:
\nA\u00a0non-terminating, non-repeating\u00a0decimal\u00a0is a\u00a0decimal\u00a0number that continues endlessly, with no group of digits repeating endlessly.\u00a0Decimals\u00a0of this type\u00a0cannot be represented as fractions, and as a result, are irrational numbers<\/p>\n

Focus on math practices<\/strong>
\nConstruct Arguments is 0.12112111211112… a rational number? Explain.
\nAnswer:
\n0.12112111211112… can’t be represented in the form of \\(\\frac{p}{q}\\) and it has non terminating non-repeating decimal expansion
\nHence, from the above,
\nWe can conclude that 0.12112111211112… is not a rational number<\/p>\n

? Essential Question<\/strong>
\nHow is an irrational number different from a rational number?
\nAnswer:
\nNumbers that can be expressed in \\(\\frac{a}{b}\\) or fraction form are rational numbers where a is an integer and b is a non-zero integer and the irrational numbers are the numbers that cannot be written in \\(\\frac{a}{b}\\) form<\/p>\n

Try It!<\/strong>
\nClassify each number as rational or irrational.
\n\"Envision
\n\"Envision
\nAnswer:
\nThe given numbers are:
\n\"Envision
\nNow,
\nWe know that,
\nNumbers that can be expressed in \\(\\frac{a}{b}\\) or fraction form are rational numbers where a is an integer and b is a non-zero integer and the irrational numbers are the numbers that cannot be written in \\(\\frac{a}{b}\\) form
\nHence,
\nThe representation of the rational and irrational numbers are:
\n\"\"<\/p>\n

Convince Me!<\/strong>
\nConstruct Arguments Jen classifies the number 4.567 as irrational because it does not repeat. Is Jen correct? Explain.
\nAnswer:
\nThe given decimal is: 4.567
\nWe know that,
\nA rational number is a number that can be written in the form of \\(\\frac{a}{b}\\)
\nA terminating decimal has the finite number of digits without repeating and it is also a rational number
\nSo,
\nWe can observe that we can write 4.567 as a rational number
\nHence, from the above,
\nWe can conclude that Jen is not correct<\/p>\n

Try It!<\/strong>
\nClassify each number as rational or irrational and explain.
\nA) \\(\\frac{2}{3}\\)
\nB) \\(\\sqrt{25}\\)
\nC) -0.7\\(\\overline{5}\\)
\nD) \\(\\sqrt{2}\\)
\nE) 7,548,123
\nAnswer:
\nThe given numbers are:
\nA) \\(\\frac{2}{3}\\)
\nB) \\(\\sqrt{25}\\)
\nC) -0.7\\(\\overline{5}\\)
\nD) \\(\\sqrt{2}\\)
\nE) 7,548,123
\nNow,
\nWe know that,
\nA “Rational number” is a number that can be written in the form of \\(\\frac{a}{b}\\)
\nEx: Terminating decimals, perfect squares, etc
\nAn “Irrational number” is a number that can not be written in the form of \\(\\frac{a}{b}\\)
\nEx: Non-terminating decimals
\nSo,
\nFrom the given numbers,
\nRational numbers ——> A, B, E
\nIrrational numbers ——> C, D<\/p>\n

KEY CONCEPT<\/strong>
\nNumbers that are not rational are called irrational numbers.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How is an irrational number different from a rational number?
\nAnswer:
\nNumbers that can be expressed in \\(\\frac{a}{b}\\) or fraction form are rational numbers where a is an integer and b is a non-zero integer and the irrational numbers are the numbers that cannot be written in \\(\\frac{a}{b}\\) form<\/p>\n

Question 2.
\nReasoning<\/strong> How can you tell whether a square root of a whole number is rational or irrational?
\nAnswer:
\nIf the square root of an integer is itself an integer (Ex: \u221a4 = 2), then by definition it is rational – If the square root is not an integer (Ex: \u221a2 = 1.41414…), then it must be irrational. Put another way the only integers for which the square root of an integer can be rational is if is a perfect square – that is where x is an integer<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Could a number ever be both rational and irrational? Explain.
\nAnswer:
\nNo. A\u00a0rational number\u00a0is a\u00a0number\u00a0that\u00a0can\u00a0be expressed as the quotient of two integers. An\u00a0irrational number\u00a0is a\u00a0number\u00a0that cannot be expressed as a quotient of two integers. So if a\u00a0number\u00a0is either\u00a0rational\u00a0or\u00a0irrational, it cannot also be the other.<\/p>\n

Do You Know How?<\/p>\n

Question 4.
\nIs the number 65.4349224… rational or irrational? Explain.
\nAnswer:
\nThe given number is 65.4349224…
\nFrom the given number,
\nWe can observe that the given number is a non-repeating and non-terminating decimal number
\nHence, from the above,
\nWe can conclude that the given number is an irrational number<\/p>\n

Question 5.
\nIs the number \\(\\sqrt{2,500}\\) rational or irrational? Explain.
\nAnswer:
\nThe given number is: \\(\\sqrt{2,500}\\)
\nWe know that,
\nA perfect square number is a rational number
\nSo,
\n\\(\\sqrt{2,500}\\) = 50
\nHence, from the above,
\nWe can conclude that the given number is a rational number<\/p>\n

Question 6.
\nClassify each number as rational or irrational.
\n\"Envision
\nAnswer:
\nWe know that,
\nA “Rational number” is a number that can be written in the form of \\(\\frac{a}{b}\\)
\nEx: Perfect squares
\nAn “Irrational number” is a number that can not be written in the form of \\(\\frac{a}{b}\\)
\nEx: Non-terminating decimal numbers
\nHence,
\nThe representation of the rational and irrational numbers from the given numbers are:
\n\"\"<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nIs 5.787787778… a rational or irrational number? Explain.
\nAnswer:
\nThe given number is 5.787787778…
\nFrom the given number,
\nWe can observe that the given number is a non-repeating and non-terminating decimal number
\nHence, from the above,
\nWe can conclude that the given number is an irrational number<\/p>\n

Question 8.
\nIs \\(\\sqrt{42}\\) rational or irrational? Explain.
\nAnswer:
\nThe given number is \\(\\sqrt{42}\\)
\nFrom the given number,
\nWe can observe that the given number is not a perfect square
\nHence, from the above,
\nWe can conclude that the given number is an irrational number<\/p>\n

Question 9.
\nA teacher places seven cards, lettered A-G, on a table. Which cards show irrational numbers?
\n\"Envision
\nAnswer:
\nThe given cards are:
\n\"Envision
\nWe know that,
\nAn “Irrational number” is a number that can not be written in the form of \\(\\frac{a}{b}\\)
\nHence, from the above,
\nWe can conclude that from the given cards,
\nThe irrational numbers are:
\nA) \u03c0
\nB) 8.25635…,
\nC) 6.\\(\\overline{31}\\)<\/p>\n

Question 10.
\nCircle the irrational number in the list below.
\nA) 7.\\(\\overline{27}\\)
\nB) \\(\\frac{5}{9}\\)
\nC) \\(\\sqrt{15}\\)
\nD) \\(\\sqrt{196}\\)
\nAnswer:
\nThe given numbers are:
\nA) 7.\\(\\overline{27}\\)
\nB) \\(\\frac{5}{9}\\)
\nC) \\(\\sqrt{15}\\)
\nD) \\(\\sqrt{196}\\)
\nNow,
\nWe know that,
\nA “Rational number” is a number that can be written in the form of \\(\\frac{a}{b}\\)
\nEx: Perfect squares
\nAn “Irrational number” is a number that can not be written in the form of \\(\\frac{a}{b}\\)
\nEx: Non-terminating decimal numbers
\nHence, from the above,
\nWe can conclude that
\nFrom the given numbers,
\nThe irrational numbers are A) and C)<\/p>\n

Question 11.
\nLisa writes the following list of numbers.
\n5.737737773…, 26, \\(\\sqrt{45}\\), –\\(\\frac{3}{2}\\), 0, 9
\nAnswer:
\nThe given numbers are:
\nA) 5.7377377737… B) 26\u00a0 C) \\(\\sqrt{45}\\)
\nD) –\\(\\frac{3}{2}\\) E) 0 F) 9
\nNow,
\nWe know that,
\nA “Rational number” is a number that can be written in the form of \\(\\frac{a}{b}\\)
\nEx: Perfect squares
\nAn “Irrational number” is a number that can not be written in the form of \\(\\frac{a}{b}\\)
\nEx: Non-terminating decimal numbers<\/p>\n

a. Which numbers are rational?
\nAnswer:
\nFrom the given numbers,
\nThe rational numbers are: B, D, E, and F<\/p>\n

b. Which numbers are irrational?
\nAnswer:
\nFrom the given numbers,
\nThe irrational numbers are: A and C<\/p>\n

Question 12.
\nConstruct Arguments<\/strong> Deena says that 9.565565556… is a rational number because it has a repeating pattern. Do you agree? Explain.
\nAnswer:
\nThe given number is 9.565565556…
\nFrom the given number,
\nWe can observe that the number is a non-repeating and a non-terminating decimal
\nSo,
\nThe given number is an irrational number
\nHence, from the above,
\nWe can conclude that we don’t have to agree with Deena<\/p>\n

Question 13.
\nIs \\(\\sqrt{1,815}\\) rational? Explain.
\nAnswer:
\nThe given number is: \\(\\sqrt{1,815}\\)
\nWe know that,
\nA “Rational number” is a number that can be written in the form of \\(\\frac{a}{b}\\)
\nEx: Perfect squares
\nNow,
\nFrom the given square root,
\nWe can observe that it won’t form a perfect square
\nHence, from the above,
\nWe can conclude that the given number is an irrational number<\/p>\n

Question 14.
\nIs the decimal form of \\(\\frac{13}{3}\\) Explain.
\nAnswer:
\nThe given number is: \\(\\frac{13}{3}\\)
\nWe know that,
\n\\(\\frac{13}{3}\\) = 4.3333…..
\nWe know that,
\nAn “Irrational number” is a number that can not be written in the form of \\(\\frac{a}{b}\\)
\nHence, from the above,
\nWe can conclude that the decimal form of \\(\\frac{13}{3}\\) is an irrational number<\/p>\n

Question 15.
\nWrite the side length of the square rug as a square root. Is the side length a rational or irrational number? Explain.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nFrom the given figure,
\nWe can observe that the given figure depicts the shape of a square
\nNow,
\nLet the side length of a square be x
\nWe know that,
\nArea = (Side length)\u00b2
\nx\u00b2 = 100
\nx = \\(\\sqrt{100}\\)
\nWe know that,
\nA “Perfect square” is a rational number
\nHence, from the above,
\nWe can conclude that a side length is a rational number<\/p>\n

Question 16.
\nReasoning<\/strong> The numbers 2.888… and 2.999… are both rational numbers. What is an irrational number that is between the two rational numbers?
\nAnswer:
\nA rational number is a number which is can be represented as the quotient of two numbers without having any remainder i.e., having remainder 0. For example 2.45, 2, 3 etc.
\nAn irrational number has a non-zero remainder and has a nonterminating quotient.
\nHence,
\nThe numbers between 2.888… and 2.999… are 2.8889………, 2.8890…….., 2.8891…… etc<\/p>\n

Question 17.
\nHigher Order Thinking<\/strong> You are given the expressions \\(\\sqrt{76+n}\\) and \\(\\sqrt{2 n+26}\\). What is the smallest value of n that will make each number rational?
\nAnswer:
\nThe given expressions are: \\(\\sqrt{76+n}\\) and \\(\\sqrt{2 n+26}\\)
\nNow,
\nTo find the smallest value of n so that each expression will be a rational number,
\n\\(\\sqrt{76+n}\\) = \\(\\sqrt{2 n+26}\\)
\nSquaring on both sides
\nSo,
\n76 + n = 2n + 26
\n2n – n = 76 – 26
\nn = 50
\nHence, from the above,
\nWe can conclude that the smallest value of n so that the given expressions will become a rational number is: 5<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 18.
\nWhich numbers are rational?
\nI. 1.1111111…
\nII. 1.567
\nIII. 1.101101110…
\nA. II and III
\nB. III only
\nC. II only
\nD. I and II
\nE I only
\nF. None of the above
\nAnswer:
\nThe given numbers are:
\nI. 1.1111111…
\nII. 1.567
\nIII. 1.101101110…
\nWe know that,
\nA “Rational number” is a number that can be written in the form of \\(\\frac{a}{b}\\)
\nSo,
\nFrom the given numbers,
\n1 and 2 are the rational numbers
\nHence, from the above,
\nwe can conclude that option D matches with the given situation<\/p>\n

Question 19.
\nDetermine whether the following numbers are rational or irrational.
\n\"Envision
\nAnswer:
\nThe representation of the given numbers as rational and irrational numbers is:
\n\"\"<\/p>\n

Lesson 1.3 Compare and Order Real Numbers<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Courtney and Malik are buying a rug to fit in a 50-square-foot space. Which rug should they purchase? Explain.<\/p>\n

\"Envision
\nAnswer:
\nIt is given that
\nCourtney and Malik are buying a rug to fit in a 50-square-foot space
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the above figure,
\nWe can observe that
\nThe rugs are in different shapes i.e., a square, a circle, and a rectangle
\nNow,
\nThe area of a square rug = The side length of a square rug \u00d7 The side length of a square rug
\n= 7 \u00d7 7
\n= 49 ft\u00b2
\nNow,
\nThe area of a circular rug = \u03c0 \u00d7 \\(\\frac{Diameter of a circular rug\u00b2}{4}\\)
\n= 3.14 \u00d7 \\(\\frac{8 \u00d78}{4}\\)
\n= 3.14 \u00d716
\n= 50.24 ft\u00b2
\nNow,
\nThe area of a rectangular rug = Length \u00d7 Width
\n= 6 \u00d7 8\\(\\frac{1}{2}\\)
\n= 6 \u00d7 \\(\\frac{17}{2}\\)
\n= \\(\\frac{6 \u00d717}{2}\\)
\n= 51 ft\u00b2
\nNow,
\nWhen we compare the area of the rugs,
\nThe area of the square rug is less than 50 ft\u00b2
\nHence, from the above,
\nWe can conclude that
\nCourtney and Malik should buy the square rug<\/p>\n

Focus on math practices<\/strong>
\nMake Sense and Persevere<\/strong> How did you decide which rug Courtney and Malik should purchase?
\nAnswer:
\nIt is given that
\nCourtney and Malik are buying a rug to fit in a 50 ft\u00b2 space
\nSo,
\nTo fit in a 50 ft\u00b2 space,
\nThe area of any type of rug should be less than 50 ft\u00b2
\nNow,
\nFrom the above problem,
\nWe can observe that
\nThe area of the square rug is the only area that is less than 50 ft\u00b2
\nHence, from the above,
\nWe can conclude that
\nCourtney and Mali should purchase the rugs based on the areas of the rugs<\/p>\n

?Essential Question<\/strong>
\nHow can you compare and order rational and irrational numbers?
\nAnswer:
\nIn the given numbers, one of them is rational while other one is irrational. To make the comparison,\u00a0let us first make the given irrational number into rational number and then carry out the comparison. So, let us square both the given numbers<\/p>\n

Try It!<\/strong>
\nBetween which two whole numbers is \\(\\sqrt{12}\\)?
\n\"Envision
\nAnswer:
\nThe given number is: \\(\\sqrt{12}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\sqrt{12}\\) is in between 3 and 4<\/p>\n

Convince Me!<\/strong>
\nWhich of the two numbers is a better estimate for \\(\\sqrt{12}\\)? Explain.
\nAnswer:
\nThe given number is: \\(\\sqrt{12}\\)
\nNow,
\nWe know that,
\n12 will be 3\u00b2 and 4\u00b2
\nSo,
\n\\(\\sqrt{12}\\) will be between 3 and 4
\nNow,
\nWhen we observe the numbers between 3 and 4
\nThe value of \\(\\frac{12}\\) will be near to 3.4
\nHence, from the above,
\nWe can conclude that
\nThe two numbers that are better estimate for \\(\\sqrt{12}\\) is: 3 and 4<\/p>\n

Try It!<\/strong>
\nCompare and order the following numbers:
\n\\(\\sqrt{11}\\), 2\\(\\frac{1}{4}\\), -2.5, 3.\\(\\overline{6}\\), -3.97621 …
\nAnswer:
\nThe given numbers are:
\n\\(\\sqrt{11}\\), 2\\(\\frac{1}{4}\\), -2.5, 3.\\(\\overline{6}\\), -3.97621 …
\nSo,
\n\\(\\sqrt{11}\\) \u2245 3.3
\n2\\(\\frac{1}{4}\\) = 2.25
\n3.\\(\\overline{6}\\) = 3.666……
\nSo,
\nThe order of the numbers from the least to the greatest is:
\n-3.97621…….. < -2.5 < 2.25 < \\(\\sqrt{11}\\) < 3.\\(\\overline{6}\\)<\/p>\n

KEY CONCEPT<\/strong>
\nTo compare rational and irrational numbers, you must first find rational approximations of the irrational numbers. You can approximate irrational numbers using perfect squares or by rounding.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question <\/strong>How can you compare and order rational and irrational numbers?
\nAnswer:
\nIn the given numbers, one of them is rational while other one is irrational. To make the comparison,\u00a0let us first make the given irrational number into rational number and then carry out the comparison. So, let us square both the given numbers<\/p>\n

Question 2.
\nReasoning<\/strong>
\nThe “leech” is a technical term for the slanted edge of a sail. Is the length of the leech shown closer to 5 meters or 6 meters? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe “leech” is a technical term for the slanted edge of a sail
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe length of the leech is: \\(\\sqrt{30}\\) meters
\nNow,
\nWe know that,
\n5\u00b2 < 30 < 6\u00b2
\n5 < \\(\\sqrt{30}\\) < 6
\nNow,
\nWe know that,
\n\\(\\sqrt{30}\\) \u2245 5.4
\nSo,
\n\\(\\sqrt{30}\\) is close to 5
\nHence, from the above,
\nWe can conclude that
\nThe length of the leech shown above is close to 5 meters<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> which is a better approximation of \\(\\sqrt{20}\\), 4.5 or 4.47? Explain.
\nAnswer:
\nThe given number is: \\(\\sqrt{20}\\)
\nNow,
\nWe know that,
\n4\u00b2 < 20 < 5\u00b2
\n4 < \\(\\sqrt{20}\\) < 5
\nNow,
\nWe know that,
\n4.5\u00b2 = 20.25
\nSo,
\nThe value of \\(\\sqrt{20}\\) is close to 4.4
\nHence, from the above,
\nWe can conclude that
\nThe better approximation of \\(\\sqrt{20}\\) is: 4.4<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nApproximate \\(\\sqrt{39}\\) to the nearest whole number.
\nAnswer:
\nThe given number is: \\(\\sqrt{39}\\)
\nNow,
\nWe know that,
\n6\u00b2 < 39 < 7\u00b2
\n6 < \\(\\sqrt{39}\\) < 7
\nNow,
\nWe know that,
\n6.5\u00b2 = 42.25
\nSo,
\nThe value of \\(\\sqrt{39}\\) is close to 6.2
\nSo,
\nThe value of \\(\\sqrt{39}\\) is closes to 6 that is the nearest whole number
\nHence, from the above,
\nWe can conclude that
\nThe better approximation of \\(\\sqrt{39}\\) that is the closest to the whole number is: 6<\/p>\n

Question 5.
\nApproximate \\(\\sqrt{18}\\) to the nearest tenth and plot the number on a number line.
\n\"Envision
\nAnswer:
\nThe given number is: \\(\\sqrt{18}\\)
\nNow,
\nWe know that,
\n4\u00b2 < 18 < 5\u00b2
\n4 < \\(\\sqrt{18}\\) < 5
\nNow,
\nWe know that,
\n4.5\u00b2 = 20.25
\n4.2\u00b2 = 17.69
\nSo,
\n\\(\\sqrt{18}\\) is close to 4.2
\nHence, from the above,
\nWe can conclude that
\nThe representation of the approximate value of \\(\\sqrt{18}\\) on the given number line is:
\n\"\"
\nThe approximate value of \\(\\sqrt{18}\\) is: 4.2<\/p>\n

Question 6.
\nCompare 5.7145… and \\(\\sqrt{29}\\). Show your work.
\nAnswer:
\nThe given numbers are: 5.7145…… and \\(\\sqrt{29}\\)
\nNow,
\nWe know that,
\n5\u00b2 < 29 < 6\u00b2
\nSo,
\n5 < \\(\\sqrt{29}\\) < 6
\nNow,
\nWe know that,
\n5.5\u00b2 = 30.25
\n5.3\u00b2 = 28.09
\nSo,
\nThe approximate value of \\(\\sqrt{29}\\) is: 5.3
\nNow,
\nWhen we compare the given numbers,
\nWe can observ ethat
\n5.7145….. > 5.3
\nHence, from the above,
\nWe can conclude that
\nThe order of the given numbers is:
\n5.7145……….. > \\(\\sqrt{29}\\)<\/p>\n

Question 7.
\nCompare and order the following numbers
\n5.2, -5.\\(\\overline{6}\\), 3\\(\\frac{9}{10}\\), \\(\\sqrt{21}\\)
\nAnswer:
\nThe given numbers are:
\n5.2, -5.\\(\\overline{6}\\), 3\\(\\frac{9}{10}\\), \\(\\sqrt{21}\\)
\nNow,
\nWe know that,
\n-5.\\(\\overline{6}\\) = -5.666666…..
\n3\\(\\frac{9}{10}\\) = 3.9
\n\\(\\sqrt{21}\\) \u2245 4.58
\nSo,
\nThe order of the given numbers from the least to the greatest is:
\n-5.\\(\\overline{6}\\) < 3\\(\\frac{9}{10}\\) < \\(\\sqrt{21}\\) < 5.2
\nHence, from the above,
\nWe can conclude that
\nThe order of the given numbers from the least to the greatest is:
\n-5.\\(\\overline{6}\\) < 3\\(\\frac{9}{10}\\) < \\(\\sqrt{21}\\) < 5.2<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 8.
\nLeveled Practice<\/strong> Find the rational approximation of \\(\\sqrt{15}\\).
\na. Approximate using perfect squares.
\n\"Envision < 15 < \"Envision
\n\"Envision = < \\(\\sqrt{15}\\) < \"Envision
\n\"Envision < \\(\\sqrt{15}\\) < \"Envision
\nAnswer:
\nThe given number is: \\(\\sqrt{15}\\)
\nNow,
\nWe know that,
\nBy using the approximation using the perfect squares,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe approximate numbers that are between \\(\\sqrt{15}\\) are: 3 and 4<\/p>\n

b. Locate and plot \\(\\sqrt{15}\\) on a number line. Find a better approximation using decimals.
\n\"Envision
\nAnswer:
\nFrom Part (a),
\nWe know that,
\nThe approximate numbers that are between \\(\\sqrt{15}\\) are: 3 and 4
\nNow,
\nWe know that,
\n3.5 \u00b2 = 12.25
\nNow,
\n\"\"
\nSo,
\nThe approximate number that is the closest to \\(\\sqrt{15}\\) is: 3.8
\nHence, from the above,
\nWe can conclude that
\nThe representation of the approximation of \\(\\sqrt{15}\\) in the given number line is:
\n\"\"
\nThe approximate number that is close to \\(\\sqrt{15}\\) is: 3.8<\/p>\n

Question 9.
\nCompare – 1.96312… and –\\(\\sqrt{5}\\). Show your work.
\nAnswer:
\nThe given numbers are: -1.96312…… and –\\(\\sqrt{5}\\)
\nNow,
\nWe know that,
\n2\u00b2 < 5 < 3\u00b2
\n2 < \\(\\sqrt{5}\\) < 3
\nNow,
\nWe know that,
\n2.5\u00b2 = 6.25
\n2.2\u00b2 = 4.84
\nSo,
\nThe approximate value of –\\(\\sqrt{5}\\) is: -2.2
\nSo,
\nThe order of the given numbers from the least to the greatest is:
\n-1.96312……. > –\\(\\sqrt{5}\\)
\nHence, from the above,
\nWe can conclude that
\nThe order of the given numbers from the least to the greatest is:
\n-1.96312……. > –\\(\\sqrt{5}\\)<\/p>\n

Question 10.
\nDoes \\(\\frac{1}{6}\\), -3, \\(\\sqrt{7}\\), –\\(\\frac{6}{5}\\), or 4.5 come first when the numbers are listed from least to greatest? Explain.
\nAnswer:
\nThe given numbers are: \\(\\frac{1}{6}\\), -3, \\(\\sqrt{7}\\), –\\(\\frac{6}{5}\\), and 4.5
\nNow,
\n\\(\\frac{1}{6}\\) = 0.166
\n\\(\\sqrt{7}\\) = 2.64
\n–\\(\\frac{6}{5}\\) = -1.2
\nSo,
\nThe order of the given numbers from the least to the greatest is:
\n-3 < –\\(\\frac{6}{5}\\) < \\(\\frac{1}{6}\\) < \\(\\sqrt{7}\\) < 4.5
\nHence, from the above,
\nWe can conclude that
\n“-3” will come first when the given numbers will be arranged from the least to the greatest<\/p>\n

Question 11.
\nA museum director wants to hang the painting on a wall. To the nearest foot, how tall does the wall need to be?
\n\"Envision
\nAnswer:
\nIt is given that
\nA museum director wants to hang the painting on a wall
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can obsere that
\nThe painting on a wall is about \\(\\sqrt{90}\\) ft
\nNow,
\nWe know that,
\n9\u00b2 < 90 < 10\u00b2
\n9 < \\(\\sqrt{90}\\) < 10
\nNow,
\nWe know that,
\n9.5\u00b2 = 90.25
\nSo,
\nThe approximate value of \\(\\sqrt{90}\\) is: 9.4 ft
\nHence, from the above,
\nWe can conclude that
\nThe height of the wall needed to hang a painting is about 9.4 ft<\/p>\n

Question 12.
\nDina has several small clay pots. She wants to display them in order of height, from shortest to tallest. What will be the order of the pots?
\n\"Envision
\nAnswer:
\nIt is given that
\nDina has several small clay pots. She wants to display them in order of height, from shortest to tallest
\nNow,
\nThe given heights are:
\n\\(\\sqrt{8}\\), 2\\(\\frac{1}{3}\\), \\(\\sqrt{5}\\), and 2.5
\nNow,
\nWe now that,
\n\\(\\sqrt{8}\\) \u2245 2.82 in.
\n2\\(\\frac{1}{3}\\) = 2.033 in.
\n\\(\\sqrt{5}\\) \u2245 2.23 in.
\nSo,
\nThe order of the heights from the shortest to the tallest is:
\n2\\(\\frac{1}{3}\\) in. < \\(\\sqrt{5}\\) in. < 2.5 in. < \\(\\sqrt{8}\\) in.
\nHence, from the above,
\nWe can conclude that
\nThe order of the pots is:
\n2\\(\\frac{1}{3}\\) in. < \\(\\sqrt{5}\\) in. < 2.5 in. < \\(\\sqrt{8}\\) in.<\/p>\n

Question 13.
\nRosie is comparing \\(\\sqrt{7}\\) and 3.44444…. She says that \\(\\sqrt{7}\\) > 3.44444… because \\(\\sqrt{7}\\) = 3.5.
\na. What is the correct comparison?
\nAnswer:
\nIt is given that
\nRosie is comparing \\(\\sqrt{7}\\) and 3.44444…. She says that \\(\\sqrt{7}\\) > 3.44444… because \\(\\sqrt{7}\\) = 3.5.
\nNow,
\nWe know that,
\n2\u00b2 < 7 < 3\u00b2
\n2 < \\(\\sqrt{7}\\) < 3
\nNow,
\nWe know that,
\n2.6\u00b2 = 6.76
\n2.7\u00b2 = 7.29
\nSo,
\nThe approximate value of \\(\\sqrt{7}\\) is: 2.6
\nSo,
\nThe order of the given numbers is:
\n\\(\\sqrt{7}\\) < 3.44444…..
\nHence, from the above,
\nWe can conclude that
\nThe correct comparison of the given numbers is:
\n\\(\\sqrt{7}\\) < 3.44444…..<\/p>\n

b. Critique Reasoning<\/strong> What mistake did Rosie likely make?
\nAnswer:
\nThe given numbers are: \\(\\sqrt{7}\\) and 3.44444…..
\nNow,
\nIt is also given that
\n\\(\\sqrt{7}\\) = 3.5
\nBut, 3.5 = \\(\\frac{7}{2}\\)
\nHence, from the above,
\nWe can conclude that
\nThe mistake Rosie likely make is:
\nShe considered \\(\\sqrt{7}\\) = \\(\\frac{7}{2}\\)<\/p>\n

Question 14.
\nModel with Math<\/strong> Approximate – \u221a23 to the nearest tenth. Draw the point on the number line.
\n\"Envision
\nAnswer:
\nThe given number is: –\\(\\sqrt{23}\\)
\nNow,
\nWe know that,
\n4\u00b2 < 23 < 5\u00b2
\n4 < \\(\\sqrt{23}\\) < 5
\nNow,
\nWe know that,
\n4.5\u00b2 = 20.25
\n4.7\u00b2 = 22.09
\nSo,
\nThe approximate value of –\\(\\sqrt{23}\\) is: -4.7
\nHence,
\nThe representtaion of the approximate value of –\\(\\sqrt{23}\\) on the given number line is:
\n\"\"<\/p>\n

Question 15.
\nHigher Order Thinking<\/strong> The length of a rectangle is twice the width. The area of the rectangle is 90 square units. Note that you can divide the rectangle into two squares.
\n\"Envision
\na. Which irrational number represents the length of each side of the squares?
\nAnswer:
\nIt is given that
\nThe length of a rectangle is twice the width. The area of the rectangle is 90 square units. Note that you can divide the rectangle into two squares.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nAccording to the given information,
\nThe area of each square = \\(\\frac{90}{2}\\)
\n= 45 square units
\nNow,
\nWe know that,
\nThe area of a square = Side\u00b2
\nSo,
\nSide of a squre = \\(\\sqrt{The area of a square}\\)
\nSo,
\nThe side length of each square = \\(\\sqrt{45}\\) units
\nHence, from the above,
\nWe can conclude that
\nThe irrational number that represents the length of each side of the squares is: \\(\\sqrt{45}\\) units<\/p>\n

b. Estimate the length and width of the rectangle.
\nAnswer:
\nIt is given that
\nThe length of a rectangle is twice the width. The area of the rectangle is 90 square units. Note that you can divide the rectangle into two squares.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nLet the width of the rectangle be x units
\nSo,
\nThe length of the rectangle = 2 (Width) = 2x units
\nNow,
\nWe know that,
\nThe length of a rectangle = Length \u00d7 Width
\nSo,
\nAccording to the given information,
\n90 = 2x \u00d7 x
\n90 = 2x\u00b2
\nx\u00b2 = \\(\\frac{90}{2}\\)
\nx\u00a0= \\(\\sqrt{45}\\)
\nHence, from the above,
\nWe can conclude that
\nThe length of the rectangle is: 2\\(\\sqrt{45}\\) units
\nThe width of the rectangle is: \\(\\sqrt{45}\\) units<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 16.
\nWhich list shows the numbers in order from least to greatest?
\nA. -4, –\\(\\frac{9}{4}\\), \\(\\frac{1}{2}\\), 3.7, \\(\\sqrt{5}\\)
\nB. -4, –\\(\\frac{9}{4}\\), \\(\\frac{1}{2}\\), \\(\\sqrt{5}\\), 3.7
\nC. –\\(\\frac{9}{4}\\), \\(\\frac{1}{2}\\), 3.7, \\(\\sqrt{5}\\), -4
\nD. –\\(\\frac{9}{4}\\), -4, \\(\\frac{1}{2}\\), 3.7, \\(\\sqrt{5}\\)
\nAnswer:
\nThe given list of numbers are:
\n-4, –\\(\\frac{9}{4}\\), \\(\\frac{1}{2}\\), 3.7, \\(\\sqrt{5}\\)
\nNow,
\nWe know that,
\n–\\(\\frac{9}{4}\\) = -2.25
\n\\(\\frac{1}{2}\\) = 0.5
\n\\(\\sqrt{5}\\) \u2245 2.23
\nSo,
\nThe order of the given list of numbers from the least to the greatest is:
\n-4 < –\\(\\frac{9}{4}\\) < \\(\\frac{1}{2}\\) < \\(\\sqrt{5}\\) < 3.7
\nHence, from the above,
\nWe can conclude that
\nThe list that shows the numbers from the least to the greatest is:
\n\"\"<\/p>\n

Question 17.
\nThe area of a square poster is 31 square inches. Find the length of one side of the poster. Explain.<\/p>\n

PART A<\/strong>
\nTo the nearest whole inch
\nAnswer:
\nIt is given that
\nThe area of a square poster is 31 square inches
\nNow,
\nWe know that,
\nThe area of a square = Side\u00b2
\nSide = \\(\\sqrt{Area of a square}\\)
\nSo,
\nAccording to the given information,
\nThe length of one side of the poster = \\(\\sqrt{31}\\) inches
\nNow,
\nWe know that,
\n5\u00b2 < 31 < 6\u00b2
\n5 < \\(\\sqrt{31}\\) < 6
\nNow,
\nWe know that,
\n5.5\u00b2 = 30.25
\n5.6\u00b2 = 31.36
\nSo,
\nThe approximate value of \\(\\sqrt{31}\\) is: 5.5
\nHence, from the above,
\nWe can conclude that
\nThe length of one side of the poster to the nearest whole inch is: 6 inches<\/p>\n

PART B<\/strong>
\nTo the nearest tenth of an inch
\nAnswer:
\nIt is given that
\nThe area of a square poster is 31 square inches
\nNow,
\nWe know that,
\nThe area of a square = Side\u00b2
\nSide = \\(\\sqrt{Area of a square}\\)
\nSo,
\nAccording to the given information,
\nThe length of one side of the poster = \\(\\sqrt{31}\\) inches
\nNow,
\nWe know that,
\n5\u00b2 < 31 < 6\u00b2
\n5 < \\(\\sqrt{31}\\) < 6
\nNow,
\nWe know that,
\n5.5\u00b2 = 30.25
\n5.6\u00b2 = 31.36
\nSo,
\nThe approximate value of \\(\\sqrt{31}\\) is: 5.5
\nHence, from the above,
\nWe can conclude that
\nThe length of one side of the poster to the nearest tenth of an inch is: 5.5 inches<\/p>\n

Lesson 1.4 Evaluate Square Roots and Cube Roots<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

ACTIVITY<\/strong><\/p>\n

Matt and his dad are building a tree house. They buy enough flooring material to cover an area of 36 square feet. What are all possible dimensions of the floor?
\n\"Envision<\/p>\n

Answer:
\nIt is given that
\nMatt and his dad are building a tree house. They buy enough flooring material to cover an area of 36 square feet
\nNow,
\nTo find the length and width of the floor, find the multiples of 36
\nSo,
\nThe multiples of 36 are:
\n36 = 1 \u00d7 36, 2 \u00d7 18, 3 \u00d7 12, 4 \u00d7 9, 9 \u00d74, 12 \u00d7 3, 18 \u00d7 2, 36 \u00d7 1
\nSo,
\nAll the possible dimensions of the floor are:
\n1 \u00d7 36, 2 \u00d7 18, 3 \u00d7 12, 4 \u00d7 9, 9 \u00d74, 12 \u00d7 3, 18 \u00d7 2, 36 \u00d7 1
\nHence, from the above,
\nWe can conclude that
\nAll the possible dimensions of the floor are:
\n1 \u00d7 36, 2 \u00d7 18, 3 \u00d7 12, 4 \u00d7 9, 9 \u00d74, 12 \u00d7 3, 18 \u00d7 2, 36 \u00d7 1<\/p>\n

Look for Relationships<\/strong>
\nCan different floor dimensions result in the same area?
\nAnswer:
\nFrom the above problem,
\nWe can observe that
\nAll the possible dimensions of the floor are:
\n1 \u00d7 36, 2 \u00d7 18, 3 \u00d7 12, 4 \u00d7 9, 9 \u00d74, 12 \u00d7 3, 18 \u00d7 2, 36 \u00d7 1
\nNow,
\nWhen we find the area by using all the different dimensions of the floor,
\nWe can observe that the area of the floor is the same
\nHence, from the above
\nWe can conclude that
\nThe different floor dimensions result in the same area<\/p>\n

Focus on math practices<\/strong>
\nReasoning<\/strong> Why is there only one set of dimensions for a square floor when there are more sets for a rectangular floor? Are all the dimensions reasonable? Explain.
\nAnswer:
\nFrom the above problem,
\nWe can observe that
\nAll the possible dimensions of the floor are:
\n1 \u00d7 36, 2 \u00d7 18, 3 \u00d7 12, 4 \u00d7 9, 9 \u00d74, 12 \u00d7 3, 18 \u00d7 2, 36 \u00d7 1
\nNow,
\nWe know that,
\nA square has the same side lengths
\nA square has the same parallel side lengths
\nHence,
\nThe square floor has only one set of dimensions whereas the rectangular floor has more sets and all the dimensions will be reasonable<\/p>\n

? Essential Question<\/strong>
\nHow do you evaluate cube roots and square roots?
\nAnswer:
\nLet the number be: p
\nNow,
\nThe square of a number is: p\u00b2
\nThe cube of a number is: p\u00b3
\nThe square root of a number is: \\(\\sqrt{p}\\)
\nThe cube root of a number is: \\(\\sqrt[3]{p}\\)<\/p>\n

Try It!<\/strong>
\nA cube-shaped art sculpture has a volume of 64 cubic feet. What is the length of each edge of the cube?
\nThe length of each edge is \"Envision feet.
\n\"Envision
\nAnswer:
\nIt is given that
\nA cube-shaped art sculpture has a volume of 64 cubic feet.
\nNow,
\nWe know that,
\nA cube has all the same side lengths
\nNow,
\nLet the side length of a cube be: s
\nSo,
\nThe volume of a cube (V) = s\u00b3
\nSo,
\nSide = \\(\\sqrt[3]{V}\\)
\nNow,
\n\"\"
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of each edge of the cube is: 4 feet<\/p>\n

Convince Me!<\/strong>
\nHow can you find the cube root of 64?
\nAnswer:
\nLet the number be: p
\nNow,
\nThe cube root of a number is: \\(\\sqrt[3]{p}\\)
\nSo,
\nThe cube root of 64 = \\(\\sqrt[3]{64}\\)
\n= \\(\\sqrt[3]{4 \u00d7 4 \u00d7 4}\\)
\n= 4
\nHence, from the above,
\nWe can conclude that
\nThe cube root of 64 is: 4<\/p>\n

Try It!<\/strong><\/p>\n

Evaluate.
\na. \\(\\sqrt[3]{27}\\)
\nAnswer:
\nThe given number is: \\(\\sqrt[3]{27}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe cube root of the given number is: 3<\/p>\n

b. \\(\\sqrt{25}\\)
\nAnswer:
\nThe given number is: \\(\\sqrt{25}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe square root of the given number is: 5<\/p>\n

c. \\(\\sqrt{81}\\)
\nAnswer:
\nThe given number is: \\(\\sqrt{81}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe square root of the given number is: 9<\/p>\n

d. \\(\\sqrt[3]{1}\\)
\nAnswer:
\nThe given number is: \\(\\sqrt[3]{1}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe cube root of the given number is: 1<\/p>\n

Try It!<\/strong><\/p>\n

Emily wants to buy a tablecloth to cover a square card table. She knows the tabletop has an area of 9 square feet. What are the minimum dimensions of the tablecloth Emily needs?
\n\"Envision
\nEmily should buy a tablecloth that measures at least
\n\"Envision feet by \"Envision feet.
\nAnswer:
\nIt is given that
\nEmily wants to buy a tablecloth to cover a square card table. She knows the tabletop has an area of 9 square feet.
\nNow,
\nWe know that,
\nThe area of a square = Side\u00b2
\nSo,
\nAccording to the given information,
\nThe area of a square table = Side\u00b2
\nSide\u00b2 = 9
\nNow,
\n\"\"
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe minimum dimensions of the table cloth Emily needs is: 3 feet \u00d7 3 feet<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

The cube root of a number is a number whose cube is equal to that number.
\n\"Envision
\nCubing a number and taking the cube root of the number are inverse operations.<\/p>\n

The square root of a number is a number whose square is equal to that number.
\n\"Envision
\nSquaring a number and taking the square root of the number are inverse operations.<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How do you evaluate cube roots and square roots?
\nAnswer:
\nLet the number be: p
\nNow,
\nThe square of a number is: p\u00b2
\nThe cube of a number is: p\u00b3
\nThe square root of a number is: \\(\\sqrt{p}\\)
\nThe cube root of a number is: \\(\\sqrt[3]{p}\\)<\/p>\n

Question 2.
\nGeneralize<\/strong> A certain number is both a perfect square and a perfect cube. Will its square root and its cube root always be different numbers? Explain.
\nAnswer:
\nWe know that,
\nA perfect square is a number whose square root is an integer; and a perfect cube is a number whose cube root is an integer.
\nA number that is a perfect square and perfect cube will not always have different numbers as its square root and cube root.<\/p>\n

Question 3.
\nCritique Reasoning<\/strong> A cube-shaped box has a volume of 27 cubic inches. Bethany says each side of the cube measures 9 inches because 9 \u00d7 3 = 27. Is Bethany correct? Explain your reasoning.
\nAnswer:
\nIt is given that
\nA cube-shaped box has a volume of 27 cubic inches. Bethany says each side of the cube measures 9 inches because 9 \u00d7 3 = 27
\nNow,
\nWe know that,
\nA “Cube” has the equal side lengths
\nNow,
\nWe know that,
\nThe volume of a cube (V) = Side\u00b3
\nSo,
\nSide = \\(\\sqrt[3]{V}\\)
\nNow,
\nFor the volume of 27 cubic inches,
\nSide = \\(\\sqrt[3]{27}\\)
\n= \\(\\sqrt[3]{3 \u00d7 3 \u00d73}\\)
\n= 3 inches
\nSo,
\nEach side of the cube measures 3 inches
\nHence, from the above,
\nWe can conclude that
\nBethany is not correct<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nA cube has a volume of 8 cubic inches. What is the length of each edge of the cube?
\nAnswer:
\nIt is given that
\nA cube has a volume of 8 cubic inches
\nNow,
\nWe know that,
\nA “Cube” has the equal side lengths
\nNow,
\nWe know that,
\nThe volume of a cube (V) = Side\u00b3
\nSo,
\nSide = \\(\\sqrt[3]{V}\\)
\nNow,
\nFor the volume of 8 cubic inches,
\nSide = \\(\\sqrt[3]{8}\\)
\n= \\(\\sqrt[3]{2 \u00d7 2 \u00d72}\\)
\n= 2 inches
\nHence, from the above,
\nWe can conclude that
\nThe length of each edge of the given cube is: 2 inches<\/p>\n

Question 5.
\nBelow is a model of the infield of a baseball stadium. How long is each side of the infield?
\n\"Envision
\nAnswer:
\nIt is given that
\nBelow is a model of the infield of a baseball stadium
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe infield is in the form of a square
\nNow,
\nWe know that,
\nThe area of a square = Side\u00b2
\nSo,
\nEach Side of the infield =\\(\\sqrt{The area of the infield}\\)
\n= \\(\\sqrt{81}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of each side of the infield is: 9 inches<\/p>\n

Question 6.
\nJulio cubes a number and then takes the cube root of the result. He ends up with 20. What number did Julio start with?
\nAnswer:
\nIt is given that
\nJulio cubes a number and then takes the cube root of the result. He ends up with 20
\nNow,
\nLet us say that the number is x.
\nSo
\nThe first step that Julio do is to cube the number so that the number will become: x\u00b3
\nThen,
\nThe next he did was take cube root of the number, so that the result will become: \\(\\sqrt[3]{x\u00b3}\\)
\nNow,
\nBy solving the above expression,
\n\\(\\sqrt[3]{x\u00b3}\\) = x
\nNow,
\nIt is given that
\nThe end result is 20
\nSo,
\nx = 20
\nSo,
\nJulio started and ended with the same number which is 20.
\nHence, from the above,
\nWe can conclude that
\nThe number did Julio start with is: 20<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice<\/strong>
\nIn 7 and 8, evaluate the cube root or square root.<\/strong><\/p>\n

Question 7.
\nRelate the volume of the cube to the length
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"\"
\nNow,
\nWe know that,
\nThe volume of a cube (V) = Side\u00b3
\nSo,
\nSide = \\(\\sqrt[3]{V}\\)
\nSo,
\nSide = \\(\\sqrt[3]{8}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of each edge of the cube is: 2 cm<\/p>\n

Question 8.
\nRelate the area of the square to the length of each edge.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"\"
\nNow,
\nWe know that,
\nThe area of a square (V) = Side\u00b2
\nSo,
\nSide = \\(\\sqrt{V}\\)
\nSo,
\nSide = \\(\\sqrt{16}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of each side of the square is: 4 cm<\/p>\n

Question 9.
\nWould you classify the number 169 as a perfect square, a perfect cube, both, or neither? Explain.
\nAnswer:
\nThe given number is: 169
\nNow,
\nWe know that,
\nA perfect cube is a number that can be expressed as the product of three equal integers
\nA perfect square is a number that can be expressed as the product of two equal integers
\nNow,
\n169 can be written as:
\n169 = 13 \u00d7 13
\nHence, from the above,
\nWe can conclude that
\n169 would be classified as a perfect square<\/p>\n

Question 10.
\nThe volume of a cube is 512 cubic inches. What is the length of each side of the cube?
\nAnswer:
\nIt is given that
\nThe volume of a cube is 512 cubic inches
\nNow,
\nWe know that,
\nThe volume of a cube (V) = Side\u00b3
\nSo,
\nSide = \\(\\sqrt[3]{V}\\)
\nSo,
\nSide = \\(\\sqrt[3]{512}\\)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of each edge of the cube is: 8 inches<\/p>\n

Question 11.
\nA square technology chip has an area of 25 square cm. How long is each side of the chip?
\nAnswer:
\nIt is given that
\nA square technology chip has an area of 25 square cm.
\nNow,
\nWe know that,
\nThe area of a square = Side\u00b2
\nSo,
\nSide = \\(\\sqrt{The area of a square}\\)
\nSo,
\nSide = \\(\\sqrt{25}\\)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of each side of the chip is: 5 cm<\/p>\n

Question 12.
\nWould you classify the number 200 as a perfect square, a perfect cube, both, or neither? Explain.
\nAnswer:
\nThe given number is: 200
\nNow,
\nWe know that,
\nA perfect cube is a number that can be expressed as the product of three equal integers
\nA perfect square is a number that can be expressed as the product of two equal integers
\nNow,
\n200 can be written as:
\n200 = 100 \u00d7 2
\n= 10 \u00d7 10 \u00d7 2
\nHence, from the above,
\nWe can conclude that
\n200 would not be classified neither as a perfect square nor a perfect cube<\/p>\n

Question 13.
\nA company is making building blocks. What is the length of each side of the block?
\n\"Envision
\nAnswer:
\nIt is given that
\nA company is making building blocks
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe building blocks is in the form of a cube
\nNow,
\nWe know that,
\nThe volume of a cube (V) = Side\u00b3
\nSo,
\nSide = \\(\\sqrt[3]{V}\\)
\nSo,
\nSide = \\(\\sqrt[3]{1}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of each side of the block is: 1 ft<\/p>\n

Question 14.
\nMrs. Drew wants to build a square sandbox with an area of 121 square feet. What is the total length of wood Mrs. Drew needs to make the sides of the sandbox?
\nAnswer:
\nIt is given that
\nMrs. Drew wants to build a square sandbox with an area of 121 square feet
\nNow,
\nWe know that,
\nThe area of a square (A) = Side\u00b2
\nSo,
\nSide = \\(\\sqrt{A}\\)
\nSo,
\nSide = \\(\\sqrt{121}\\)
\nNow,
\n\"\"
\nSo,
\nThe side of the sandbox is: 11 feet
\nNow,
\nTo find the total length of wood Mrs. Drew needs to make the sides of the sandbox = 4 \u00d7 (The length of the side of the sandbox)
\n= 4 \u00d7 11
\n= 44 feet
\nHence, from the above,
\nWe can conclude that
\nThe total length of wood Mrs. Drew needs to make the sides of the sandbox is: 44 feet<\/p>\n

Question 15.
\nConstruct Arguments<\/strong> Diego says that if you cube the number 4 and then take the cube root of the result, you end up with 8. Is Diego correct? Explain.
\nAnswer:
\nIt is given that
\nDiego says that if you cube the number 4 and then take the cube root of the result, you end up with 8
\nNow,
\nAccording to the given information,
\nStep 1:
\n4\u00b3 = 4 \u00d7 4 \u00d7 4
\n= 64
\nStep 2:
\n\\(\\sqrt[3]{64}\\)
\n= \\(\\sqrt[3]{4 \u00d7 4 \u00d74}\\)
\n= 4
\nBut,
\nIt is given that
\nThe end result is 8 and we got 4
\nHence,f rom the above,
\nWe can conclude that
\nDiego is not correct<\/p>\n

Question 16.
\nHigher Order Thinking<\/strong> Talia is packing a moving box. She has a square-framed poster with an area of 9 square feet. The cube-shaped box has a volume of 30 cubic feet. Will the poster lie flat in the box? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nTalia is packing a moving box. She has a square-framed poster with an area of 9 square feet. The cube-shaped box has a volume of 30 cubic feet
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nTo make the square-framed poster fit into a cube-shaped box,
\nThe side of square-framed poster < The side of each edge of the cube-shaped box
\nNow,
\nWe know that,
\nThe area of a square = Side\u00b2
\nThe volume of a cube = Side\u00b3
\nSo,
\n\\(\\sqrt{9}\\) = 3 feet
\n\\(\\sqrt[3]{30}\\) = 3.10 feet
\nSo,
\n3 < 3.10
\nSo,
\nThe side of square-framed poster < The side of each edge of the cube-shaped box
\nHence, from the above,
\nWe can conclude that
\nThe poster lie flat in the box<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 17.
\nWhich expression has the greatest value?
\nA. \\(\\sqrt{49}\\) . 2
\nB. \\(\\sqrt{49}\\) – \\(\\sqrt{16}\\)
\nC. \\(\\sqrt{25}\\) + \\(\\sqrt{16}\\)
\nD. \\(\\sqrt{25}\\).3
\nAnswer:
\nThe given expressions are:
\na.
\nThe given expression is: \\(\\sqrt{49}\\) . 2
\nNow,
\nWe know that,
\n\\(\\sqrt{49}\\) = 7
\nSo,
\nThe value of the given expression is: 14
\nb.
\nThe given expression is:
\n\\(\\sqrt{49}\\) – \\(\\sqrt{16}\\)
\nNow,
\nWe know that,
\n\\(\\sqrt{49}\\) = 7
\n\\(\\sqrt{16}\\) = 4
\nSo,
\nThe value of the given expression is: 3
\nc.
\nThe given expression is:
\n\\(\\sqrt{25}\\) + \\(\\sqrt{16}\\)
\nNow,
\nWe know that,
\n\\(\\sqrt{25}\\) = 5
\n\\(\\sqrt{16}\\) = 4
\nSo,
\nThe value of the given expression is: 9
\nd.
\nThe given expression is:
\n\\(\\sqrt{25}\\).3
\nNow,
\nWe know that,
\n\\(\\sqrt{25}\\) = 5
\nSo,
\nThe value of the given expression is: 15
\nHence, from the above,
\nWe can conclude that
\nThe expression that has the greatest value is:
\n\"\"<\/p>\n

Question 18.
\nA toy has various shaped objects that a child can push through matching holes. The area of the square hole is 8 square cm. The volume of a cube-shaped block is 64 cubic cm
\nPART A<\/strong>
\nWhich edge length can you find? Explain.
\nAnswer:
\nIt is given that
\nA toy has various shaped objects that a child can push through matching holes. The area of the square hole is 8 square cm. The volume of a cube-shaped block is 64 cubic cm
\nNow,
\nWe know that,
\nThe area of a square (A) = Side\u00b2
\nThe volume of a cube (V) = Side\u00b3
\nSo,
\n\\(\\sqrt{8}\\) = 2.82 cm
\n\\(\\sqrt[3]{64}\\) = 4 cm
\nHence, from the above,
\nWe can conclude that
\nThe value of the edge lengths you found are:
\nThe side of a square-shaped hole is: 2.82 cm
\nThe side of a cube-shaped block is: 4 cm<\/p>\n

PART B<\/strong>
\nWill the block fit in the square hole? Explain.
\nAnswer:
\nNow,
\nFrom Part A,
\nWe can observe that
\nThe side of a square-shaped hole is: 2.82 cm
\nThe side of a cube-shaped block is: 4 cm
\nNow,
\nFor the block to fit in the square hole,
\nThe side of the block < The side of the hole
\nBut,
\n4 cm > 2.82 cm
\nHence, from the above,
\nWe can conclude that
\nThe block will not fit in the square hole<\/p>\n

Lesson 1.5 Solve Equations Using Square Roots and Cube Roots<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Janine can use up to 150 one-inch blocks to build a solid, cube-shaped model. What are the dimensions of the possible models that she can build? How many blocks would Janine use for each model? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nJanine can use up to 150 one-inch blocks to build a solid, cube-shaped model
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe volume of a cube = Length \u00d7 Width \u00d7 Height
\nNow,
\nTo find the dimensions of the possible models that Janine can model,
\nWe have to find the multiples of 150 in terms of three
\nSo,
\n150 = 25 \u00d7 6
\n150 = 5 \u00d7 5 \u00d7 6
\nSo,
\nThe total number of blocks Janine would use for each model = The sum of the above three multiples of 150
\n= 5 + 5 + 6
\n= 16 blocks
\nHence, from the above,
\nWe can conclude that
\nThe dimensions of the possible model that Janine can build is: 5 \u00d7 5 \u00d7 6
\nThe total number of blocks Janine would use for each model is: 16 blocks<\/p>\n

Look for Relationships<\/strong>
\nHow are the dimensions of a solid related to its volume?
\nAnswer:
\nThe volume, V , of any rectangular solid is the product of the length, width, and height. We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, B , is equal to length \u00d7 Width.<\/p>\n

Focus on math practices<\/strong>
\nReasoning<\/strong> Janine wants to build a model using \\(\\frac{1}{2}\\)-inch cubes. How many \\(\\frac{1}{2}\\)-inch cubes would she use to build a solid, cube-shaped model with side lengths of 4 inches? Show your work.
\nAnswer:
\nIt is given that
\nJanine wants to build a model using \\(\\frac{1}{2}\\)-inch cubes and a cube-shaped model with side lengths of 4 inches
\nNow,
\nAccording to the given information,
\nThe number of \\(\\frac{1}{2}\\)-inch cubes would Janine used to build a solid = \\(\\frac{1}{2}\\) \u00d7 8 blocks
\nHence, from the above,
\nWe can conclude that
\nThe number of \\(\\frac{1}{2}\\)-inch cubes would she use to build a solid, cube-shaped model with side lengths of 4 inches is: 8 blocks<\/p>\n

? Essential Question<\/strong>
\nHow can you solve equations with squares and cubes?
\nAnswer:
\nThe steps to solve equations with squares are:
\nStep 1:
\nDivide all terms by a (the coefficient of x2<\/sup>).
\nStep 2:
\nMove the number term (c\/a) to the right side of the equation.
\nStep 3:
\nComplete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.
\nThe steps to solve equations with cubes are:
\nStep 1:
\nDivide all terms by a (the coefficient of x\u00b3<\/sup>).
\nStep 2:
\nMove the number term (\\(\\frac{d}{a}\\)) to the right side of the equation.
\nStep 3:
\nComplete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.<\/p>\n

Try It!<\/strong><\/p>\n

What is the side length, s, of the square below?
\n\"Envision<\/p>\n

Each side of the square measures \"Envision meters.
\nAnswer:
\nThe given figure is:
\n\"\"
\nNow,
\nFrom the given figure,
\nWe can observe that
\nIt is a square
\nNow,
\nWe know that,
\nThe area of a square = Side\u00b2
\nNow,
\n\"\"
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nEach side of the square measures 10 meters<\/p>\n

Convince Me!<\/strong>
\nWhy are there two possible solutions to the equation s2<\/sup> = 100? Explain why only one of the solutions is valid in this situation.
\nAnswer:
\nThe given equation is: s\u00b2
\nNow,
\nWe know that,
\nThe square of a positive number or a negative number is always positive and the square root of a number must always be positive
\nSo,
\ns\u00b2 = 100
\ns = \\(\\sqrt{100}\\)
\ns = \u00b1 10
\nNow,
\nWe know that,
\nThe side of any figure will always be positive
\nHence, from the above,
\nWe can conclude that
\nOnly one of the solutions is valid in this situation because of the property of sides of the geometrical figures<\/p>\n

Try It!<\/strong><\/p>\n

Solve x3<\/sup> = 64.
\nAnswer:
\nThe given equation is:
\nx\u00b3 = 64
\nSo,
\nx = \\(\\sqrt[3]{64}\\)
\nx = \\(\\sqrt[3]{4 \u00d7 4 \u00d7 4}\\)
\nx = 4
\nHence, from the above,
\nWe can conclude that
\nThe value of x for the given equation is: 4<\/p>\n

Try It!<\/strong><\/p>\n

a. Solve a3<\/sup> = 11.
\nAnswer:
\nThe given equation is:
\na\u00b3 = 11
\nSo,
\na = \\(\\sqrt[3]{11}\\)
\nNow,
\nWe know that,
\nThe cube of a number will always be positive
\nHence, from the above,
\nWe can conclude that
\nThe possible solution for the given equation is: \\(\\sqrt[3]{11}\\)<\/p>\n

b. Solve c2<\/sup> = 27.
\nAnswer:
\nThe given equation is:
\nc\u00b2 = 27
\nSo,
\nc = \u00b1\\(\\sqrt{27}\\)
\nHence, from the above,
\nWe can conclude that
\nThe possible solutions for the given equation is: \\(\\sqrt{27}\\), –\\(\\sqrt{27}\\)<\/p>\n

KEY CONCEPT<\/strong>
\nYou can use square roots to solve equations involving squares.
\nx2 <\/sup> = a
\n\\(\\sqrt{x^{2}}\\) = \\(\\sqrt{a}\\)
\nx = + \\(\\sqrt{a}\\), –\\(\\sqrt{a}\\)<\/p>\n

You can use cube roots to solve equations involving cubes. x2 = b Vx3 = xb
\nx3<\/sup> = b
\n\\(\\sqrt[3]{x^{3}}\\) = \\(\\sqrt[3]{b}\\)
\nx = \\(\\sqrt[3]{b}\\)<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong>
\nHow can you solve equations with squares and cubes?
\nAnswer:
\nThe steps to solve equations with squares are:
\nStep 1:
\nDivide all terms by a (the coefficient of x2<\/sup>).
\nStep 2:
\nMove the number term (c\/a) to the right side of the equation.
\nStep 3:
\nComplete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.
\nThe steps to solve equations with cubes are:
\nStep 1:
\nDivide all terms by a (the coefficient of x\u00b3<\/sup>).
\nStep 2:
\nMove the number term (\\(\\frac{d}{a}\\)) to the right side of the equation.
\nStep 3:
\nComplete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.<\/p>\n

Question 2.
\nBe Precise<\/strong> Suri solved the equation x2<\/sup> = 49 and found that x = 7. What error did Suri make?
\nAnswer:
\nIt is given that
\nSuri solved the equation x2<\/sup> = 49 and found that x = 7.
\nNow,
\nThe given equation is:
\nx\u00b2 = 49
\nNow,
\nWe know that,
\nThe square of a number is always positive but the square root of a number can either be positive or negative
\nSo,
\nx = \\(\\sqrt{49}\\)
\nx = \u00b17
\nx = 7, -7
\nHence, from the above,
\nWe can conclude that
\nThe error did Suri made is that he did not consider the negative square root of 49<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> There is an error in the work shown below. Explain the error and provide a correct solution.
\nx3<\/sup> = 125
\n\\(\\sqrt[3]{x^{3}}\\) = \\(\\sqrt[3]{125}\\)
\nx = 5 and x = -5
\nAnswer:
\nThe given equation is:
\nx\u00b3 = 125
\nNow,
\nWe know that,
\nThe cube root of a number will always be positive
\nSo,
\n\\(\\sqrt[3]{x^{3}}\\) = \\(\\sqrt[3]{125}\\)
\nx = \\(\\sqrt[3]{x^{3}}\\) = \\(\\sqrt[3]{5 \u00d7 5 \u00d7 5}\\)
\nx = 5
\nHence, from the above,
\nWe can conclude that
\nThe error is not considering that the cube root of a number will always be positive
\nThe correct solution is:
\nx = 5<\/p>\n

Question 4.
\nWhy are the solutions to x2<\/sup> = 17 irrational?
\nAnswer:
\nThe given equation is:
\nx\u00b2 = 17
\nSo,
\nx = \u00b1\\(\\sqrt{17}\\)
\nNow,
\nWe know that,
\nThe perfect square will be an integer and we know that an integer is a rational number
\nNow,
\nWhen we look at 17,
\nIt is not a perfect square
\nHence, from the above,
\nWe can conclude that
\nThe solutions of the given equation are irrational<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 5.
\nIf a cube has a volume of 27 cubic cm, what is the length of each edge? Use the volume formula, V = s3<\/sup>, and show your work.
\nAnswer:
\nIt is given that
\nA cube has a volume of 27 cubic cm
\nNow,
\nWe know that,
\nThe volume of a cube (V) = Side\u00b3
\nSo,
\nSide = \\(\\sqrt[3]{V}\\)
\nNow,
\nAccording to the given information,
\nSide = \\(\\sqrt[3]{27}\\)
\nSide = \\(\\sqrt[3]{3 \u00d7 3 \u00d7 3}\\)
\nSide = 3 cm
\nHence, from the above,
\nWe can conclude that
\nThe length of each edge of the give cube is: 3 cm<\/p>\n

Question 6.
\nDarius is building a square launch pad for a rocket project. If the area of the launch pad is 121 square cm, what is its side length? Use the area formula, A = s2<\/sup>, and show your work.
\n\"Envision
\nAnswer:
\nIt is given that
\nDarius is building a square launch pad for a rocket project. If the area of the launch pad is 121 square cm,
\nNow,
\nWe know that,
\nThe area of a square (A) = Side\u00b2
\nSo,
\nSide = \\(\\sqrt{A}\\)
\nNow,
\nAccording to the given information,
\nSide = \\(\\sqrt{121}\\)
\nSide = \\(\\sqrt{11 \u00d7 11}\\)
\nSide = 11 cm
\nHence, from the above,
\nWe can conclude that
\nThe side length of the launch pad is: 1 cm<\/p>\n

Question 7.
\nSolve the equation x3<\/sup> = -215.
\nAnswer:
\nThe given equation is:
\nx\u00b3 = -215
\nNow,
\nWe know that,
\nThe square root won’t accept negative values but the cube root will accept both positive and negative values
\nSo,
\nx = \\(\\sqrt[3]{-215}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe solution for the given equation is: -5.99<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice<\/strong>
\nIn 8 and 9, solve.<\/strong><\/p>\n

Question 8.
\nz2<\/sup> = 1
\n\"Envision
\nAnswer:
\nThe given equation is:
\nz\u00b2 = 1
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

Question 9.
\na3<\/sup> = 216
\n\"Envision
\nAnswer:
\nThe given equation is:
\na\u00b3 = 216
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe solution for the given equation is: 6<\/p>\n

Question 10.
\nSolve v2<\/sup> = 47.
\nAnswer:
\nThe give equation is:
\nv\u00b2 = 47
\nNow,
\n\\(\\sqrt{v\u00b2}\\) = \\(\\sqrt{47}\\)
\nv = \u00b1\\(\\sqrt{47}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe solutions for the given equation are: 6.85, and -6.85<\/p>\n

Question 11.
\nThe area of a square photo is 9 square inches. How long is each side of the photo?
\nAnswer:
\nIt is given that
\nThe area of a square photo is 9 square inches
\nNow,
\nWe know that,
\nThe area of a square (A) = Side\u00b2 (s)
\nSo,
\ns\u00b2 = 9
\n\\(\\sqrt{s\u00b2}\\) = \\(\\sqrt{9}\\)
\ns = 3 inches [Because the length of the side will never be negative]
\nHence, from the above,
\nWe can conclude that
\nThe length of each side of the photo is: 3 inches<\/p>\n

Question 12.
\nSolve the equation y2<\/sup> = 81.
\nAnswer:
\nThe given equation is:
\ny\u00b2 = 81
\nNow,
\n\\(\\sqrt{y\u00b2}\\) = \u00b1\\(\\sqrt{81}\\)
\ny = \u00b19
\nHence, from the above,
\nWe can conclude that
\nThe solutions for the given equation are: 9, and -9<\/p>\n

Question 13.
\nSolve the equation w3<\/sup> = 1,000.
\nAnswer:
\nThe given equation is:
\nw\u00b3 = 1,000
\nNow,
\n\\(\\sqrt[3]{w\u00b3}\\) = \\(\\sqrt[3]{1,000}\\)
\nw = \\(\\sqrt[3]{10 \u00d7 10 \u00d7 10}\\)
\nw = 10
\nHence, from the above,
\nWe can conclude that
\nThe solution for the given equation is: 10<\/p>\n

Question 14.
\nThe area of a square garden is shown. How long is each side of the garden?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe area of a square garden is shown.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe area of the square garden is: 121 ft\u00b2
\nNow,
\nWe know that,
\nThe area of a square (A) = s\u00b2
\nSo,
\nAccording to the given information,
\ns\u00b2 = 121
\n\\(\\sqrt{s\u00b2}\\) = \\(\\sqrt{121}\\)
\ns = 11 ft [Since the length of the side will never be negative]
\nHence, from the above,
\nWe can conclude that
\nThe length of each side of the garden is: 11 ft<\/p>\n

Question 15.
\nSolve b2<\/sup> = 77.
\nAnswer:
\nThe given equation is:
\nb\u00b2 = 77
\nNow,
\n\\(\\sqrt{b\u00b2}\\) = \u00b1\\(\\sqrt{77}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe solutions for the givene quation are: 8.77, and -8.77<\/p>\n

Question 16.
\nFind the value of c in the equation c3<\/sup> = 1,728.
\nAnswer:
\nThe given equation is:
\nc\u00b3 = 1,728
\nNow,
\n\\(\\sqrt[3]{c\u00b3}\\) = \\(\\sqrt[3]{1,728}\\)
\nc = 12
\nHence, from the above,
\nWe can conclude that
\nThe value of c for the given equation is: 12<\/p>\n

Question 17.
\nSolve the equation v3<\/sup> = 12.
\nAnswer:
\nThe given equation is:
\nv\u00b3 = 12
\nNow,
\n\\(\\sqrt[3]{v\u00b3}\\) = \\(\\sqrt[3]{12}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe solution for the given equation is: 2.28<\/p>\n

Question 18.
\nHigher Order Thinking<\/strong> Explain why
\n\\(\\sqrt[3]{-\\frac{8}{27}}\\) is –\\(\\frac{2}{3}\\)
\nAnswer:
\nThe given equation is:
\n\\(\\sqrt[3]{-\\frac{8}{27}}\\)
\nNow,
\nWe know that,
\n\\(\\sqrt[3]{\\frac{a}{b}}\\) = \\(\\frac{\\sqrt[3]{a}}{\\sqrt[3]{b}}\\)
\nSo,
\n\\(\\sqrt[3]{-\\frac{8}{27}}\\) = –\\(\\frac{\\sqrt[8]{a}}{\\sqrt[3]{27}}\\)
\n= –\\(\\frac{\\sqrt[2 \u00d7 2 \u00d72]{a}}{\\sqrt[3]{3 \u00d73 \u00d7 3}}\\)
\n= –\\(\\frac{2}{3}\\)
\nHence, from the above,
\nWe can conclude that
\n\\(\\sqrt[3]{-\\frac{8}{27}}\\) is –\\(\\frac{2}{3}\\) due to the below property of Exponents:
\n\\(\\sqrt[3]{\\frac{a}{b}}\\) = \\(\\frac{\\sqrt[3]{a}}{\\sqrt[3]{b}}\\)<\/p>\n

Question 19.
\nCritique Reasoning<\/strong> Manolo says that the solution of the equation g2<\/sup> = 36 is g = 6 because 6 \u00d7 6 = 36. Is Manolo’s reasoning complete? Explain.
\nAnswer:
\nIt is given that
\nManolo says that the solution of the equation g2<\/sup> = 36 is g = 6 because 6 \u00d7 6 = 36
\nNow,
\nThe given equation is:
\ng\u00b2 = 36
\nNow,
\nWe know that,
\nThe square of a number will always be positive but the square root of a number will either be positive or negative
\nSo,
\n\\(\\sqrt{g\u00b2}\\) = \u00b1\\(\\sqrt{36}\\)
\ng = \u00b16
\nSo,
\nThe solutions for the given equation are: 6, -6
\nHence, from the above,
\nWe can conclude that
\nManolo’s reasoning is not complete<\/p>\n

Question 20.
\nEvaluate \\(\\sqrt[3]{-512}\\).
\na. Write your answer as an integer.
\nAnswer:
\nThe given equation is:
\n\\(\\sqrt[3]{-512}\\)
\nNow,
\n\\(\\sqrt[3]{-512}\\) = \\(\\sqrt[3]{(-8) \u00d7 (-8) \u00d7 (-8)}\\)
\n= -8
\nHence, from the above,
\nWe can conclude that
\nThe value of \\(\\sqrt[3]{-512}\\) as an integer is: -8<\/p>\n

b. Explain how you can check that your result is correct.
\nAnswer:
\nThe given equation is:
\n\\(\\sqrt[3]{-512}\\)
\nNow,
\nFrom part (a),
\nWe get the value of the given equation is: 8
\nNow,
\n\\(\\sqrt[3]{(-8) \u00d7 (-8) \u00d7 (-8)}\\)
\n= \\(\\sqrt[3]{64 \u00d7 (-8)}\\)
\n= \\(\\sqrt[3]{-512}\\)
\nHence, from the above,
\nWe can conclude that
\nThe result is correct because the givene quation and the result are the same<\/p>\n

Question 21.
\nYael has a square-shaped garage with 228 square feet of floor space. She plans to build an addition that will increase the floor space by 50%. What will be the length, to the nearest tenth, of one side of the new garage?
\n\"Envision
\nAnswer:
\nIt is given that
\nYael has a square-shaped garage with 228 square feet of floor space. She plans to build an addition that will increase the floor space by 50%.
\nNow,
\nAccording to the given information,
\n50% of 228 = \\(\\frac{50}{100}\\) \u00d7 228
\n= \\(\\frac{50 \u00d7 228}{100}\\)
\n= 114 square feet
\nSo,
\nThe area of the new garage = 228 + 114
\n= 342 square feet
\nNow,
\nWe know that,
\nThe area of a square (A) = Side (s)\u00b2
\nSo,
\ns\u00b2 = 342
\n\\(\\sqrt{s\u00b2}\\) = \\(\\sqrt{342}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of one side of the new garage is: 18.5 feet<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 22.
\nThe Traverses are adding a new room to their house. The room will be a cube with a volume of 6,859 cubic feet. They are going to put in hardwood floors, which costs $10 per square foot. How much will the hardwood floors cost?
\nAnswer:
\nIt is given that
\nThe Traverses are adding a new room to their house. The room will be a cube with a volume of 6,859 cubic feet. They are going to put in hardwood floors, which costs $10 per square foot
\nNow,
\nWe know that,
\nThe volume of a cube (V) = s\u00b3
\nSo,
\nAccording to the given information,
\ns\u00b3 = 6,859
\n\\(\\sqrt[3]{s\u00b3}\\) = \\(\\sqrt[3]{6,859}\\)
\ns = 19 feet
\nSo,
\nThe length of each edge of the new room is: 19 feet
\nNow,
\nTo find the total cost of hardwood floors, find the perimeter of the room and multiply the result with the cost per square foot
\nNow,
\nWe know that,
\nThe perimeter of a cube = 6s
\nSo,
\nThe perimeter of the cube (p) = 6 \u00d7 19
\n= 114 feet
\nNow,
\nThe total cost of hardwood floors = 114 \u00d7 $10
\n= $1,140
\nHence, from the above,
\nWe can conclude that
\nThe total cost of hardwood floors is: $1,140<\/p>\n

Question 23.
\nWhile packing for their cross-country move, the Chen family uses a crate that has the shape of a cube.
\nPART A<\/strong>
\nIf the crate has the volume V = 64 cubic feet, what is the length of one edge?
\nIt is given that
\nWhile packing for their cross-country move, the Chen family uses a crate that has the shape of a cube and the crate has the volume V = 64 cubic feet
\nNow,
\nWe know that,
\nThe volume of a cube (V) = s\u00b3
\nSo,
\nAccording to the given information,
\ns\u00b3 = 64
\n\\(\\sqrt[3]{s\u00b3}\\) = \\(\\sqrt[3]{64}\\)
\ns = 4 feet
\nHence, from the above,
\nWe can conclude that
\nThe length of each edge of the crate is: 4 feet<\/p>\n

PART B<\/strong>
\nThe Chens want to pack a large, framed painting. If the framed painting has the shape of a square with an area of 12 square feet, will the painting fit flat against a side of the crate? Explain.
\nAnswer:
\nIt is given that
\nThe Chens want to pack a large, framed painting. If the framed painting has the shape of a square with an area of 12 square feet
\nNow,
\nWe know that,
\nThe area of a square (A) = s\u00b2
\nSo,
\nAccording to the given information,
\ns\u00b2 = 12
\nNow,
\n\\(\\sqrt{s\u00b2}\\) = \\(\\sqrt{12}\\)
\nNow,
\n\"\"
\nSo,
\nAccording to the given information,
\nThe side of the crate > The side of the painting
\n4 > 2.28
\nHence, from the above,
\nWe can conclude that
\nThe painting will fit flat against a side of the crate<\/p>\n

Topic 1 MID-TOPIC CHECKPOINT<\/h3>\n

Question 1.
\nVocabulary<\/strong> How can you show that a number is a rational number? Lesson 1.2
\nAnswer:
\nA rational number is a number that\u00a0can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers<\/p>\n

Question 2.
\nWhich shows 0.2\\(\\overline{3}\\) as a fraction? Lesson 1.1
\nA. \\(\\frac{2}{33}\\)
\nB. \\(\\frac{7}{33}\\)
\nC. \\(\\frac{23}{99}\\)
\nD. \\(\\frac{7}{30}\\)
\nAnswer:
\nThe given expression is: 0.2\\(\\overline{3}\\)
\nHence, from the above,
\nWe can conclude that
\nThe options that show 0.2\\(\\overline{3}[\/latex as a fraction is:
\n\"\"<\/p>\n

Question 3.
\nApproximate [latex]\\sqrt{8}\\) to the nearest hundredth. Show your work. Lesson 1.3
\nAnswer:
\nThe given expression is: \\(\\sqrt{8}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe approximate value of the given expression to the nearest hundredth is: 2.82<\/p>\n

Question 4.
\nSolve the equation m2<\/sup> = 14. Lesson 1.5
\nAnswer:
\nThe given equation is:
\nm\u00b2 = 14
\nNow,
\n\\(\\sqrt{m\u00b2}\\) = \u00b1\\(\\sqrt{14}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that<\/p>\n

The solutions for the given equation are: 3.74, -3.74<\/p>\n

Question 5.
\nA fish tank is in the shape of a cube. Its volume is 125 ft3<\/sup>. What is the area of one face of the tank? Lessons 1.4 and 1.5
\nAnswer:
\nIt is given that
\nA fish tank is in the shape of a cube. Its volume is 125 ft3<\/sup>.
\nNow,
\nWe know that,
\nThe volume of a cube (V) = s\u00b3
\nSo,
\ns\u00b3= 125
\nNow,
\n\\(\\sqrt[3]{s\u00b3}\\) = \\(\\sqrt[3]{125}\\)
\ns = \\(\\sqrt[3]{5 \u00d7 5 \u00d7 5}\\)
\ns = 5 ft
\nNow,
\nWe know that,
\nThe surface area of a cube = 4s\u00b2
\nSo,
\nThe area of one face of the tank = 4 \u00d7 5\u00b2
\n= 4 \u00d7 25
\n= 100 ft\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe area of one face of the tank is: 100 ft\u00b2<\/p>\n

Question 6.
\nWrite 1.\\(\\overline{12}\\) as a mixed number. Show your work. Lesson 1.1
\nAnswer:
\nThe given expression is: 1.\\(\\overline{12}\\)
\nNow,
\nThe representation of the given expression in the form of a fraction is: \\(\\frac{28}{25}\\)
\nNow,
\nThe representation of \\(\\frac{28}{25}\\) into a mixed number is: 1\\(\\frac{3}{25}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of 1.\\(\\overline{12}\\) as a mixed number is: 1\\(\\frac{3}{25}\\)
\nHow well did you do on the mid-topic checkpoint? Fill in the stars. \"Envision<\/p>\n

Topic 1 MID-TOPIC PERFORMANCE TASK<\/h3>\n

Six members of the math club are forming two teams for a contest. The teams will be determined by having each student draw a number from a box.
\n\"Envision<\/p>\n

PART A<\/strong><\/p>\n

The table shows the results of the draw. The students who drew rational numbers will form the team called the Tigers. The students who drew irrational numbers will form the team called the Lions. List the members of each team.
\nAnswer:
\nIt is given that
\nSix members of the math club are forming two teams for a contest. The teams will be determined by having each student draw a number from a box.
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nWe know that,
\nThe numbers that can be written in the form of \\(\\frac{p}{q}\\) are “Rational numbers”
\nThe numbers that can not be written in the form of \\(\\frac{p}{q}\\) are “Irrational numbers”
\nNow,
\nFrom the given table,
\nThe list of rational numbers are:
\n6.\\(\\overline{34}\\), \\(\\sqrt{36}\\), 6.3\\(\\overline{4}\\)
\nThe list of Irrational numbers are:
\n\\(\\sqrt{38}\\), 6.343443444….,, \\(\\sqrt{34}\\)
\nHence, from the above,
\nWe can conclude that
\nThe students present in the Tigers Team are:
\n6.\\(\\overline{34}\\), \\(\\sqrt{36}\\), 6.3\\(\\overline{4}\\)
\nThe students present in the Lions Team are:
\n\\(\\sqrt{38}\\), 6.343443444….,, \\(\\sqrt{34}\\)<\/p>\n

PART B<\/strong><\/p>\n

The student on each team who drew the greatest number will be the captain of that team. Who will be the captain of the Tigers? Show your work.
\nAnswer:
\nFrom Part A,
\nWe can observe that
\nThe students present in the Tigers Team are:
\n6.\\(\\overline{34}\\), \\(\\sqrt{36}\\), 6.3\\(\\overline{4}\\)
\nNow,
\nFrom the above list of numbers,
\nWe can observe that
\n6.\\(\\overline{34}\\) is the greater number
\nHence, from the above,
\nWe can conclude that
\nThe captain of the Tigers Team is: Anya<\/p>\n

PART C<\/strong><\/p>\n

Who will be the captain of the Lions? Show your work.
\nAnswer:
\nThe students present in the Lions Team are:
\n\\(\\sqrt{38}\\), 6.343443444….,, \\(\\sqrt{34}\\)
\nNow,
\nFrom the above list of numbers,
\nWe can observe that
\n6.343443444…., is the greater number
\nHence, from the above,
\nWe can conclude that
\nThe captain of the Lions Team is: Ryan<\/p>\n

Lesson 1.6 Use Properties of Integer Exponents<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

One band’s streaming video concert to benefit a global charity costs $1.00 to view.
\nThe first day, the concert got 2,187 views. The second day, it got about three times as many views. On the third day, it got 3 times as many views as on the second day. If the trend continues, how much money will the band raise on Day 7?
\n\"Envision
\nAnswer:
\nIt is given that
\nOne band’s streaming video concert to benefit a global charity costs $1.00 to view.
\nThe first day, the concert got 2,187 views. The second day, it got about three times as many views. On the third day, it got 3 times as many views as on the second day
\nSo,
\nAccording to the given information,
\nThe amount of money got on the first day of concert = (The total number of views) \u00d7 $1
\n= 2,187 \u00d7 $1
\n= $2,187
\nThe amount of money got on the second day of concert = (The amount of money got on the first day of concert) \u00d7 3
\n= $2,187 \u00d7 3
\n= $6,561
\nThe amount of money got on the third day of concert = (The amount of money got on the second day of concert) \u00d7 3
\n= $6,561 \u00d7 3
\n= $19,683
\nThe amount of money got on the fourth day of concert = (The amount of money got on the third day of concert) \u00d7 3
\n= $19,683 \u00d7 3
\n= $59,049
\nThe amount of money got on the fifth day of concert = (The amount of money got on the fourth day of concert) \u00d7 3
\n= $59,049 \u00d7 3
\n= $1,77,147
\nThe amount of money got on the sixth day of concert = (The amount of money got on the fifth day of concert) \u00d7 3
\n= $1,77,147 \u00d7 3
\n= $5,31,441
\nThe amount of money got on the seventh day of concert = (The amount of money got on the sixth day of concert) \u00d7 3
\n= $5,31,441 \u00d7 3
\n= $15,94,323
\nHence, from the above,
\nWe can conclude that
\nThe amount of money the Band raise on Day 7 is: $15,94,323<\/p>\n

Focus on math practices<\/strong>
\nUse Structure<\/strong> Use prime factorization to write an expression equivalent to the amount of money raised by the band on the last day of the week.
\nAnswer:
\nFrom the above,
\nWe can observe that
\nThe amount of money got on the last day of the wee is: $15,94,323
\nNow,
\nBy using the Prime factorisation method,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe prime factorisation of 15,94,323 is: 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3<\/p>\n

? Essential Question<\/strong>
\nHow do properties of integer exponents help you write equivalent expressions?
\nAnswer:
\nThe properties of integer exponents can be used to write equivalent expressions by combining numeric or algebraic expressions that have a common base, distributing exponents to products and quotients, and simplifying powers of powers.<\/p>\n

\n
\n
Try It!
\n<\/strong>The local zoo welcomed a newborn African elephant that weighed 34<\/sup> kg. It is expected that at adulthood, the newborn elephant will weigh approximately 34<\/sup> times as much as its birth weight. What expression represents the expected adult weight of the newborn elephant?
\nAnswer:
\nIt is given that
\nThe local zoo welcomed a newborn African elephant that weighed 34<\/sup> kg. It is expected that at adulthood, the newborn elephant will weigh approximately 34<\/sup> times as much as its birth weight
\nNow,
\nAccording to the given information,
\nThe expected adult weight of the newborn elephant = (The actual weight of the newborn elephant) \u00d7 (The number of times the expected weight will be as much as its birth weight)
\n= 34<\/sup> \u00d7 34
\n<\/sup>Now,
\nAccording to the “Product of Powers Property”,
\nWhen multiplying two powers with the same bases, add the exponents
\nSo,
\nThe expected adult weight of the newborn elephant = 34 + 4
\n<\/sup>= 38<\/sup> kg
\nHence, from the above,
\nWe can conclude that
\nThe expected adult weight of the newborn elephant is: 38<\/sup> kg<\/div>\n<\/div>\n<\/div>\n

Convince Me!<\/strong>
\nExplain why the Product of Powers Property makes mathematical sense.
\nAnswer:
\nThe Power of a Product rule states that a term raised to a power is equal to the product of its factors raised to the same power<\/p>\n

Try It!<\/strong>
\nWrite equivalent expressions using the properties of exponents.<\/strong><\/p>\n

a. (73<\/sup>)2<\/sup>
\nAnswer:
\nThe given expression is: (7\u00b3)\u00b2
\nNow,
\nWe know that,
\nThe “Power of Powers Property” states that to find the power of a power, multiply the exponents
\nSo,
\n(7\u00b3)\u00b2 = 73 \u00d7 2<\/sup>
\n= 76<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 76<\/sup><\/p>\n

b. (45<\/sup>)3<\/sup>
\nAnswer:
\nThe given expression is: (45<\/sup>)3<\/sup>
\nNow,
\nWe know that,
\nThe “Power of Powers Property” states that to find the power of a power, multiply the exponents
\nSo,
\n(45<\/sup>)3<\/sup> = 43 \u00d7 5<\/sup>
\n= 415<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 415<\/sup><\/p>\n

c. 94<\/sup> \u00d7 84<\/sup>
\nAnswer:
\nThe given expression is: 94<\/sup> \u00d7 84<\/sup>
\nNow,
\nWe know that,
\nBy using the “Power of Products Property”, when multiplying two exponential expressions with the same exponent and different powers, multiply the bases and keep the exponents the same
\nSo,
\n94<\/sup> \u00d7 84<\/sup> = (9 \u00d7 8)4<\/sup>
\n= 724<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 724<\/sup><\/p>\n

d. 89<\/sup> \u00f7 83<\/sup>
\nAnswer:
\nThe given expression is: 89<\/sup> \u00f7 83<\/sup>
\nNow,
\nWe know that,
\nThe “Quotient of powers Property” states that when dividing two exponential expressions with the same base, subtract the exponents
\nSo,
\n89<\/sup> \u00f7 83<\/sup> = 89 – 3<\/sup>
\n= 86<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 86<\/sup><\/p>\n

KEY CONCEPT<\/strong><\/p>\n

Use these properties when simplifying expressions with exponents (when a, m, and n \u2260 0).
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\nEssential Question<\/strong> How do properties of integer exponents help you write equivalent expressions?
\nAnswer:
\nThe properties of integer exponents can be used to write equivalent expressions by combining numeric or algebraic expressions that have a common base, distributing exponents to products and quotients, and simplifying powers of powers.<\/p>\n

Question 2.
\nLook for Relationships<\/strong> If you are writing an equivalent expression for 23<\/sup> \u00d7 24<\/sup>, how many times would you write 2 as a factor?
\nAnswer:
\nIt is given that
\nyou are writing an equivalent expression for 23 <\/sup> \u00d7 24
\n<\/sup>Now,
\nThe given expression is: 23 <\/sup> \u00d7 24<\/sup>
\nNow,
\nWe know that,
\nThe “Power of Powers Property” states that to find the power of a power, multiply the exponents
\nSo,
\n23 <\/sup> \u00d7 24<\/sup> = 23 + 4<\/sup>
\n= 27<\/sup>
\n= 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2
\nHence, from the above,
\nWe can conclude that
\nWe would write 2 as a factor 7 times<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Kristen wrote 58<\/sup> as an expression equivalent to (52<\/sup>)4<\/sup>. Her math partner writes 56<\/sup>. Who is correct?
\nAnswer:
\nIt is given that
\nKristen wrote 58<\/sup> as an expression equivalent to (52<\/sup>)4<\/sup>. Her math partner writes 56<\/sup>
\nNow,
\nThe given expression is: 58<\/sup>
\nNow,
\nWe know that,
\n(am<\/sup>)n<\/sup> = amn<\/sup>
\nNow,
\nAccording to the above Property,
\n58<\/sup> = 54 \u00d7 2<\/sup>
\n= (54<\/sup>)2<\/sup>
\n= (52<\/sup>)4<\/sup>
\nHence, from the above,
\nWe can conclude that
\nKristen is correct<\/p>\n

Question 4.
\nCritique Reasoning<\/strong> Tyler says that an equivalent expression for 23<\/sup> \u00d7 53<\/sup> is 109<\/sup>. Is he correct? Explain.
\nAnswer:
\nIt is given that
\nTyler says that an equivalent expression for 23<\/sup> \u00d7 53<\/sup> is 109<\/sup>
\nNow,
\nThe given expression is: 23<\/sup> \u00d7 53<\/sup>
\nNow,
\nWe know that,
\nam<\/sup> \u00d7 bm<\/sup> = (a \u00d7 b)m<\/sup>
\nSo,
\n23<\/sup> \u00d7 53<\/sup> = (2 \u00d7 5)3<\/sup>
\n= 103<\/sup>
\nSo,
\nThe equivalent expression for 23<\/sup> \u00d7 53<\/sup> is: 103<\/sup>
\nHence, from the above,
\nWe can conclude that
\nTyler is not correct<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 5.
\nWrite an equivalent expression for 712<\/sup> \u00d7 74<\/sup>.
\nAnswer:
\nThe given expression is: 712<\/sup> \u00d7 74<\/sup>
\nNow,
\nWe know that,
\nam<\/sup> \u00d7 an<\/sup> = am + n<\/sup>
\nSo,
\n712<\/sup> \u00d7 74<\/sup> = (7)12 + 4<\/sup>
\n= 716<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 716<\/sup><\/p>\n

Question 6.
\nWrite an equivalent expression for (82<\/sup>)4<\/sup>.
\nAnswer:
\nThe given expression is: (82<\/sup>)4<\/sup>
\nNow,
\nWe know that,
\n(am<\/sup>)n<\/sup> = amn<\/sup>
\nSo,
\n(82<\/sup>)4<\/sup> = 82 \u00d7 4<\/sup>
\n= 88<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 88<\/sup><\/p>\n

Question 7.
\nA billboard has the given dimensions.
\n\"Envision
\nUsing exponents, write two equivalent expressions for the area of the rectangle.
\nAnswer:
\nIt is given that
\nA billboard has the given dimensions.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe billboard is in the form of a rectangle.
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe length of the billboard is: 7\u00b2 ft
\nThe width of the billboard is: 4\u00b2 ft
\nNow,
\nWe know that,
\nWe know that,
\nam<\/sup> \u00d7 bm<\/sup> = (a \u00d7 b)m<\/sup>
\nSo,
\nThe area of the rectangle (A) = 7\u00b2 \u00d7 4\u00b2
\n= (7 \u00d7 4)\u00b2
\n= 28\u00b2 ft\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe two equivalent expressions for the area of the rectangle is: 7\u00b2 \u00d7 4\u00b2 and 28\u00b2 ft\u00b2<\/p>\n

Question 8.
\nWrite an equivalent expression for 189<\/sup> – 184<\/sup>.
\nAnswer:
\nThe given expression is: 189<\/sup> – 184<\/sup>
\nNow,
\n189<\/sup> – 184<\/sup> = 184<\/sup> (185<\/sup> – 1)
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 184<\/sup> (185<\/sup> – 1)<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice
\nIn 9-12, use the properties of exponents to write an equivalent expression for each given expression.<\/strong><\/p>\n

Question 9.
\n28 <\/sup> \u00d7 24<\/sup>
\n\"Envision
\nAnswer:
\nThe given expression is: 28 <\/sup> \u00d7 24<\/sup>
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 212<\/sup><\/p>\n

Question 10.
\n\\(\\frac{8^{7}}{8^{3}}\\)
\n\"Envision
\nAnswer:
\nThe given expression is: \\(\\frac{87<\/sup>}{83<\/sup>}\\)
\nNow,
\nWe know that,
\n\\(\\frac{am<\/sup>}{an<\/sup>}\\) = am – n<\/sup>
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 84<\/sup><\/p>\n

Question 11.
\n(34<\/sup>)5<\/sup>
\n\"Envision
\nAnswer:
\nThe given expression is: (34<\/sup>)5<\/sup>
\nNow,
\nWe know that,
\n(am<\/sup>)n<\/sup> = amn<\/sup>
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for thegiven expression is: 320<\/sup><\/p>\n

Question 12.
\n39<\/sup> \u00d7 29<\/sup>
\n\"Envision
\nAnswer:
\nThe given expression is: 39<\/sup> \u00d7 29<\/sup>
\nNow,
\nWe know that,
\nam<\/sup> \u00d7 bm<\/sup> = (a \u00d7 b)m<\/sup>
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: (3 \u00d7\u00a0 2)9<\/sup><\/p>\n

Question 13.
\na. How do you multiply powers that have the same base?
\nAnswer:
\nIf the two exponential expressions have the same base and different exponents and both are multiplying each other, then add the exponents by keeping the bases same<\/p>\n

b. How do you divide powers that have the same base?
\nAnswer:
\nIf the two exponential expressions have the same base and different exponents and both are dividing each other, then subtract the exponents by keeping the bases same<\/p>\n

c. How do you find the power of a power?
\nAnswer:
\nWhen an exponential expression contains the power of a power, we will multiply both the powers by keeping the base constant<\/p>\n

d. How do you multiply powers with different bases but the same exponent?
\nAnswer:
\nIf the two exponential expressions have the same exponent and different bases and both are multiplying each other, then multiply the bases by keeping the exponents same<\/p>\n

Question 14.
\nWhich expressions are equivalent to 211<\/sup>?
\nSelect all that apply.
\n\"Envision \\(\\frac{2^{23}}{2^{12}}\\)
\n\"Envision 27<\/sup> \u2219 24<\/sup>
\n\"Envision \\(\\frac{2^{9}}{2^{2}}\\)
\n\"Envision 22<\/sup> \u2219 29<\/sup>
\nAnswer:
\nThe given expression is: 211<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe expressions that are equivalent to 211<\/sup> are:
\n\"\"<\/p>\n

In 15-18, use the properties of exponents to write an equivalent expression for each given expression.<\/strong><\/p>\n

Question 15.
\n(44<\/sup>)3<\/sup>
\nAnswer:
\nThe given expression is: (44<\/sup>)3<\/sup>
\nNow,
\nWe know that,
\n(am<\/sup>)n<\/sup> = am \u00d7 n<\/sup>
\nSo,
\n(44<\/sup>)3<\/sup> = 44 \u00d7 3<\/sup>
\n= 412<\/sup>
\nHence, from the above,
\nWe can conclude that
\n(44<\/sup>)3<\/sup> = 412<\/sup><\/p>\n

Question 16.
\n\\(\\frac{3^{12}}{3^{3}}\\)
\nAnswer:
\nThe given expression is: \\(\\frac{3^{12}}{3^{3}}\\)
\nNow,
\nWe know that,
\n\\(\\frac{am<\/sup>}{an<\/sup>}\\) = am – n<\/sup>
\nSo,
\n\\(\\frac{3^{12}}{3^{3}}\\) = 312 – 3<\/sup>
\n= 39<\/sup>
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{3^{12}}{3^{3}}\\) = 39<\/sup><\/p>\n

Question 17.
\n45<\/sup> \u00d7 42<\/sup>
\nAnswer:
\nThe given expression is: 45<\/sup> \u00d7 42<\/sup>
\nNow,
\nWe know that,
\nam<\/sup> \u00d7 an<\/sup> = am + n<\/sup>
\nSo,
\n45<\/sup> \u00d7 42<\/sup> = 45 + 2<\/sup>
\n= 47<\/sup>
\nHence, from the above,
\nWe can conclude that
\n45<\/sup> \u00d7 42<\/sup> = 47<\/sup><\/p>\n

Question 18.
\n64<\/sup> \u00d7 24<\/sup>
\nAnswer:
\nThe given expression is: 64<\/sup> \u00d7 24<\/sup>
\nNow,
\nWe know that,
\nam<\/sup> \u00d7 bm<\/sup> = (a \u00d7 b)m<\/sup>
\nSo,
\n64<\/sup> \u00d7 24<\/sup> = (6 \u00d7 2)4<\/sup>
\n= 124<\/sup>
\nHence, from the above,
\nWe can conclude that
\n64<\/sup> \u00d7 24<\/sup> = 124<\/sup><\/p>\n

Question 19.
\nCritique Reasoning<\/strong> Alberto incorrectly stated that \\(\\frac{5^{7}}{5^{4}}\\) = 13<\/sup>. What was Alberto’s error? Explain your reasoning and find the correct answer.
\nAnswer:
\nIt is given that
\nAlberto incorrectly stated that \\(\\frac{5^{7}}{5^{4}}\\) = 13<\/sup>
\nNow,
\nThe given expression is: \\(\\frac{5^{7}}{5^{4}}\\)
\nNow,
\nWe know that,
\n\\(\\frac{am<\/sup>}{an<\/sup>}\\) = am – n><\/sup>
\nSo,
\n\\(\\frac{5^{7}}{5^{4}}\\) = 5 7 – 4<\/sup>
\n= 53<\/sup>
\nSo,
\nFrom the above,
\nWe can observe that
\nAlberto applied the “Quotient of Powers Property” incorrectly
\nHence, from the above,
\nWe can conclude that
\nAlberto’s error is: Alberto applied the “Quotient of Powers Property” incorrectly<\/p>\n

Question 20.
\nIs the expression 8 \u00d7 85<\/sup> equivalent to (8 \u00d7 8)5<\/sup>? Explain.
\nAnswer:<\/p>\n

Question 21.
\nIs the expression (32<\/sup>)-3<\/sup> equivalent to (33<\/sup>)-2<\/sup>? Explain.
\nAnswer:<\/p>\n

Question 22.
\nIs the expression 32<\/sup> \u2219 3-3<\/sup> equivalent to 33<\/sup> \u2219 3-2<\/sup>? Explain.
\nAnswer:<\/p>\n

Question 23.
\nModel with Math<\/strong> What is the width of the rectangle written as an exponential expression?
\n\"Envision
\nAnswer:<\/p>\n

Question 24.
\nSimplify the expression \\(\\left(\\left(\\frac{1}{2}\\right)^{3}\\right)^{3}\\).
\nAnswer:<\/p>\n

Question 25.
\nHigher Order Thinking<\/strong> Use a property of exponents to write (3b)5<\/sup> as a product of powers.
\nAnswer:<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 26.
\nSelect all the expressions equivalent to 45<\/sup> \u2219 410<\/sup>
\n\"Envision 45<\/sup> + 410<\/sup>
\n\"Envision 43<\/sup> \u2219 45<\/sup>
\n\"Envision 43<\/sup> \u2219 412<\/sup>
\n\"Envision 43<\/sup> + 412<\/sup>
\n\"Envision 418<\/sup> – 43<\/sup>
\n\"Envision 415<\/sup><\/p>\n

Question 27.
\nYour teacher asks the class to evaluate the expression (23<\/sup>)1<\/sup>. Your classmate gives an incorrect answer of 16.
\nPART A<\/strong>
\nEvaluate the expression.
\nPART B<\/strong>
\nWhat was the likely error?
\nA. Your classmate divided the exponents.
\nB. Your classmate multiplied the exponents.
\nC. Your classmate added the exponents.
\nD. Your classmate subtracted the exponents.
\nAnswer:<\/p>\n

Lesson 1.7 More Properties of Integer Exponents<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Explore It!<\/strong><\/p>\n

Calvin and Mike do sit-ups when they work out. They start with 64 sit-ups for the first set and do half as many each subsequent set.
\n\"Envision<\/p>\n

Look for Relationships<\/strong>
\nDetermine whether the relationship shown for Set 1 is also true for Sets 2-5.<\/p>\n

A. What representation can you use to show the relationship between the set number and the number of sit-ups?<\/p>\n

B. What conclusion can you make about the relationship between the number of sit-ups in each set?<\/p>\n

Focus on math practices<\/strong>
\nUse Structure How could you determine the number of sit-up sets Calvin and Mike do?<\/p>\n

? Essential Question<\/strong>
\nWhat do the Zero Exponent and Negative Exponent Properties mean?<\/p>\n

Try It!<\/strong><\/p>\n

Evaluate
\na. (-7)0<\/sup>
\nb. (43)0<\/sup>
\nc. 10<\/sup>
\nd. (0.50<\/sup>)<\/p>\n

Convince Me!<\/strong>
\nWhy is 2(70<\/sup>) = 2?<\/p>\n

Try It!<\/strong><\/p>\n

Write each expression using positive exponents.
\na. 8-2<\/sup>
\nb. 2-4<\/sup>
\nc. 3-5<\/sup>
\nAnswer:<\/p>\n

Try It!<\/strong><\/p>\n

Write each expression using positive exponents.<\/p>\n

a. \\(\\frac{1}{5^{-3}}\\)<\/p>\n

b. \\(\\frac{1}{2^{-6}}\\)<\/p>\n

KEY CONCEPT<\/strong>
\nUse these additional properties when simplifying or generating equivalent expressions with exponents (when a \u2260 0 and n \u2260 0).
\nZero Exponent Property
\na0<\/sup> = 1<\/p>\n

Negative Exponent Property
\na-n<\/sup> = \\(\\frac{1}{a^{n}}\\)<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\nEssential Question<\/strong> What do the Zero Exponent and Negative Exponent Properties mean?<\/p>\n

Question 2.
\nReasoning<\/strong> In the expression 9-12<\/sup>, what does the negative exponent mean?<\/p>\n

Question 3.
\nReasoning<\/strong> in the expression 3(20<\/sup>), what is the order of operations? Explain how you would evaluate the expression.<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nSimplify 1,999,9990<\/sup>.
\nAnswer:<\/p>\n

Question 5.
\na. Write 7-6<\/sup> using a positive exponent.
\nb. Rewrite \\(\\frac{1}{10^{-3}}\\) using a positive exponent.
\nAnswer:<\/p>\n

Question 6.
\nEvaluate 27xo<\/sup>y-2<\/sup> for x = 4 and y = 3.
\nAnswer:<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice<\/strong> In 7-8, complete each table to find the value of a nonzero number raised to the power of 0.<\/p>\n

Question 7.
\n\"Envision
\nAnswer:<\/p>\n

Question 8.
\n\"Envision
\nAnswer:<\/p>\n

Question 9.
\nGiven: (-3.2)0<\/sup>
\na. Simplify the given expression.
\nb. Write two expressions equivalent to the given expression. Explain why the three expressions are equivalent.
\nAnswer:<\/p>\n

Question 10.
\nSimplify each expression for x = 6.
\na. 12x0<\/sup>(x-4<\/sup>)
\nb. 14(x-2<\/sup>)<\/p>\n

In 11 and 12, compare the values using >,<, or =.<\/p>\n

Question 11.
\n3-2<\/sup> \"Envision 1
\nAnswer:<\/p>\n

Question 12.
\n\\(\\left(\\frac{1}{4}\\right)^{0}\\) \"Envision 1<\/p>\n

In 13 and 14, rewrite each expression using a positive exponent.<\/p>\n

Question 13.
\n9-4<\/sup>
\nAnswer:<\/p>\n

Question 14.
\n\\(\\frac{1}{2^{-6}}\\)
\nAnswer:<\/p>\n

Question 15.
\nGiven: 9y0<\/sup>
\na. Simplify the expression for y = 3.
\nb. Construct Arguments Will the value of the given expression vary depending on y? Explain.
\nAnswer:<\/p>\n

Question 16.
\nSimplify each expression for x = 4.
\na. -5x-4<\/sup>
\nb. 7x-3<\/sup>
\nAnswer:<\/p>\n

Question 17.
\nEvaluate each pair of expressions.
\na. (-3)-8<\/sup> and -3-8<\/sup>
\nb. (-3)-9<\/sup> and -3-9<\/sup>
\nAnswer:<\/p>\n

Question 18.
\nBe Precise<\/strong> To win a math game, Lamar has to pick a card with an expression that has a value greater than 1. The card Lamar chooses reads \\(\\left(\\frac{1}{2}\\right)^{-4}\\). Does Lamar win the game? Explain.
\n\"Envision
\nAnswer:<\/p>\n

Question 19.
\nSimplify the expression. Assume that x is nonzero. Your answer should have only positive exponents. x-10<\/sup> \u2219 x6<\/sup>
\nAnswer:<\/p>\n

Question 20.
\nHigher Order Thinking<\/strong>
\na. Is the value of the expression \\(\\left(\\frac{1}{4^{-3}}\\right)^{-2}\\) greater than 1, equal to 1, or less than 1?
\nb. If the value of the expression is greater than 1, show how you can change one sign to make the value less than 1. If the value is less than 1, show how you can change one sign to make the value greater than 1. If the value is equal to 1, show how you can make one change to make the value not equal to 1.<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 21.
\nWhich expressions are equal to 5-3<\/sup>? Select all
\nthat apply.
\n\"Envision 125
\n\"Envision 125-1<\/sup>
\n\"Envision 53<\/sup>
\n\"Envision \\(\\frac{1}{5^{3}}\\)
\n\"Envision \\(\\frac{1}{125}\\)
\nAnswer:<\/p>\n

Question 22.
\nWhich expressions have a value less than 1 when x = 4? Select all that apply.
\n\"Envision \\(\\left(\\frac{3}{x^{2}}\\right)^{0}\\)
\n\"Envision \\(\\frac{x^{0}}{3^{2}}\\)
\n\"Envision \\(\\frac{1}{6^{-x}}\\)
\n\"Envision \\(\\frac{1}{x^{-3}}\\)
\n\"Envision 3x-4<\/sup><\/p>\n

Lesson 1.8 Use Powers of 10 to Estimate Quantities<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Explain It!<\/strong><\/p>\n

Keegan and Jeff did some research and found that there are approximately 7,492,000,000,000,000,000 grains of sand on Earth. Jeff says that it is about 7 \u00d7 1015<\/sup> grains of sand. Keegan says that this is about 7 \u00d7 1018<\/sup> grains of sand.
\n\"Envision<\/p>\n

A. How might Jeff have determined his estimate? How might Keegan have determined his estimate?<\/p>\n

B. Whose estimate, Jeff’s or Keegan’s, is more logical? Explain.<\/p>\n

Focus on math practices<\/strong>
\nBe Precise Do you think the two estimates are close in value? Explain your reasoning.<\/p>\n

? Essential Question<\/strong>
\nwhen would you use a power of 10 to estimate a quantity?<\/p>\n

Try It!<\/strong><\/p>\n

Light travels 299,792,458 meters per second. Sound travels at 332 meters per second. Use a power of 10 to compare the speed of light to the speed of sound.
\n299,792,458 rounded to the greatest place value is \"Envision
\nThere are \"Envision zeros in the rounded number.
\nThe estimated speed of light is \"Envision \u00d7 10\"Envision<\/sup> meters per second.
\n3 \u00d7 10\"Envision<\/sup> > 3 \u00d7 10\"Envision<\/sup>, so the speed of light is faster than the speed of sound.<\/p>\n

322 rounded to the greatest place value is \"Envision
\nThere are \"Envision zeros in the rounded number. The estimated speed of sound is \"Envision \u00d7 10\"Envision<\/sup> meters per second.<\/p>\n

Convince Me!<\/strong>
\nCountry A has a population of 1,238,682,005 and Country B has a population of 1,106,487,394. How would you compare these populations?<\/p>\n

Try It!<\/strong><\/p>\n

There are approximately 1,020,000,000 cars in the world. The number of cars in the United States is approximately 239,800,000.
\nCompare the number of cars in the world to that in the United States.<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

You can estimate a very large or very small number by rounding the number to its greatest place value, and then writing that number as a single digit times a power of 10.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> when would you use powers of 10 to estimate a quantity?
\nAnswer:<\/p>\n

Question 2.
\nConstruct Arguments<\/strong> Kim writes an estimate for the number 0.00436 as 4 \u00d7 103<\/sup>. Explain why this cannot be correct.
\nAnswer:<\/p>\n

Question 3.
\nBe Precise<\/strong> Raquel estimated 304,900,000,000 as 3 \u00d7 108<\/sup>. What error did she make?
\nAnswer:<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nUse a single digit times a power of 10 to estimate the height of Mt. Everest to the nearest ten thousand feet.
\n\"Envision
\nAnswer:<\/p>\n

Question 5.
\nA scientist records the mass of a proton as 0.0000000000000000000000016726231 gram. Use a single digit times a power of 10 to estimate the mass.
\nAnswer:<\/p>\n

Question 6.
\nThe tanks at the Georgia Aquarium hold approximately 8.4 \u00d7 106<\/sup> gallons of water. The tanks at the Audubon Aquarium of the Americas hold about 400,000 gallons of water. Use a single digit times a power of 10 to estimate how many times greater the amount of water is at the Georgia Aquarium.
\nAnswer:<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice<\/strong> in 7-9, use powers of 10 to estimate quantities.<\/p>\n

Question 7.
\nA city has a population of 2,549,786 people. Estimate this population to the nearest million. Express your answer as the product of a single digit and a power of 10. Rounded to the nearest million, the population is about \"Envision
\nWritten as the product of a single digit and a power of ten, this number is \"Envision
\nAnswer:<\/p>\n

Question 8.
\nUse a single digit times a power of 10 to estimate the number 0.00002468. Rounded to the nearest hundred thousandth, the number is about \"Envision
\nWritten as a single digit times a power of ten, the estimate is \"Envision
\nAnswer:<\/p>\n

Question 9.
\nThe approximate circumferences of Earth and Saturn are shown. How many times greater is the circumference of Saturn than the circumference of Earth?
\nThe circumference of Saturn is
\n\"Envision
\n\"Envision
\nSaturn’s circumference is about\"Envision times greater than the circumference of Earth.
\nAnswer:<\/p>\n

Question 10.
\nEstimate 0.037854921 to the nearest hundredth. Express your answer as a single digit times a power of ten.
\nAnswer:<\/p>\n

Question 11.
\nCompare the numbers 6 \u00d7 10-6<\/sup> and 2 \u00d7 10-8<\/sup>.
\na. Which number has the greater value?
\nb. Which number has the lesser value?
\nc. How many times greater is the greater number?
\nAnswer:<\/p>\n

Question 12.
\nTaylor made $43,785 last year. Use a single digit times a power of ten to express this value rounded to the nearest ten thousand.
\nAnswer:<\/p>\n

Question 13.
\nThe length of plant cell A is 8 \u00d7 10-5<\/sup> meter. The length of plant cell B is 0.000004 meter. How many times greater is plant cell A’s length than plant cell B’s length?
\nAnswer:<\/p>\n

Question 14.
\nCritique Reasoning<\/strong> The diameter of one species of bacteria is shown. Bonnie approximates this measure as 3 \u00d7 10-11<\/sup> meter. Is she correct? Explain.
\n\"Envision
\nAnswer:<\/p>\n

Question 15.
\nThe populations of Cities A and B are 2.6 \u00d7 105<\/sup> and 1,560,000, respectively. The population of City C is twice the population of City B. The population of City C is how many times the population of City A?
\nAnswer:<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 16.
\nEarth is approximately 5 \u00d7 109<\/sup> years old. For which of these ages could this be an approximation?
\nA. 4,762,100,000 years
\nB. 48,000,000,000 years
\nC. 4.45 \u00d7 109<\/sup> years
\nD. 4.249999999 \u00d7 109<\/sup> years
\nAnswer:<\/p>\n

Question 17.
\nPART A<\/strong>
\nExpress 0.000000298 as a single digit times a power of ten rounded to the nearest ten millionth.<\/p>\n

PART B<\/strong>
\nExplain how negative powers of 10 can be helpful when writing and comparing small numbers.
\nAnswer:<\/p>\n

Lesson 1.9 Understand Scientific Notation<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Solve & Discuss It!<\/strong><\/p>\n

Scientists often write very large or very small numbers using exponents. How might a scientist write the number shown using exponents?
\n\"Envision<\/p>\n

Use Structure<\/strong>
\nHow can you use your knowledge of powers of 10 to rewrite the number?<\/p>\n

Focus on math practices<\/strong>
\nLook for Relationships What does the exponent in 1015 tell you about the value of the number?<\/p>\n

? Essential Question<\/strong>
\nWhat is scientific notation and why is it used?<\/p>\n

Try It!<\/strong>
\nThe height of Angel Falls, the tallest waterfall in the world, is 3,212 feet. How do you write this number in scientific notation?
\n\"Envision<\/p>\n

Convince Me!<\/strong>
\nWhy do very large numbers have positive exponents when written in scientific notation? Explain.<\/p>\n

Try It!<\/strong>
\nA common mechanical pencil lead measures about 0.005 meter in diameter. How can you express this measurement using scientific notation ?<\/p>\n

Try It!<\/strong><\/p>\n

Write the numbers in standard form.
\na. 9.225 \u00d7 1018<\/sup>
\nb. 6.3 \u00d7 10-8<\/sup>
\nAnswer:<\/p>\n

KEY CONCEPT<\/strong>
\nScientific notation is a way to write very large numbers or very small numbers. Scientists use scientific notation as a more efficient and convenient way of writing such numbers.
\nA number in scientific notation is the product of two factors. The first factor must be greater than or equal to 1 and less than 10. The second factor is a power of 10.
\n\"Envision
\nTo write a number in scientific notation in standard form, multiply the decimal number by the power of 10.<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n?Essential Question<\/strong> What is scientific notation and why is it used?
\nAnswer:<\/p>\n

Question 2.
\nCritique Reasoning<\/strong> Taylor states that 2,800,000 in scientific notation is 2.8 \u00d7 10-6<\/sup> because the number has six places to the right of the 2. Is Taylor’s reasoning correct?
\nAnswer:<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Sam will write 0.000032 in scientific notation. Sam thinks that the exponent of 10 will be positive. Do you agree? Construct an argument to support your response.
\nAnswer:<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nExpress 586,400,000 in scientific notation.
\nAnswer:<\/p>\n

Question 5.
\nThe genetic information of almost every living thing is stored in a tiny strand called DNA. Human DNA is 3.4 \u00d7 10-8<\/sup> meter long. Write the length in standard form.
\n\"Envision
\nAnswer:<\/p>\n

Question 6.
\nThe largest virus known to man is the Megavirus, which measures 0.00000044 meter across. Express this number in scientific notation.
\nAnswer:<\/p>\n

Question 7.
\nHow would you write the number displayed on the calculator screen in standard form?
\n\"Envision
\nAnswer:<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice In 8 and 9, write the numbers in the correct format.<\/p>\n

Question 8.
\nThe Sun is 1.5 \u00d7 108<\/sup> kilometers from Earth. 1.5 \u00d7 108<\/sup> is written as \"Envision in standard form.
\nAnswer:<\/p>\n

Question 9.
\nBrenna wants an easier way to write 0.0000000000000000587.
\n0.0000000000000000587 is written as \"Envision \u00d7 10\"Envision in scientific notation.
\nAnswer:<\/p>\n

Question 10.
\nIs 23 \u00d7 10-8<\/sup> written in scientific notation? Justify your response.
\nAnswer:<\/p>\n

Question 11.
\nIs 8.6 \u00d7 107<\/sup> written in scientific notation? Justify your response.
\nAnswer:<\/p>\n

Question 12.
\nSimone evaluates an expression using her calculator. The calculator display is shown at the right. Express the number in standard form.
\n\"Envision
\nAnswer:<\/p>\n

Question 13.
\nExpress the number 0.00001038 in scientific notation.
\nAnswer:<\/p>\n

Question 14.
\nExpress the number 80,000 in scientific notation.
\nAnswer:<\/p>\n

Question 15.
\nPeter evaluates an expression using his calculator. The calculator display is shown at the right. Express the number in standard form.
\n\"Envision
\nAnswer:<\/p>\n

Question 16.
\na. What should you do first to write 5.871 \u00d7 10-7<\/sup> in standard form?
\nb. Express the number in standard form
\nAnswer:<\/p>\n

Question 17.
\nExpress 2.58 \u00d7 10-2<\/sup> in standard form.
\nAnswer:<\/p>\n

Question 18.
\nAt a certain point, the Grand Canyon is approximately 1,600,000 centimeters across. Express this number in scientific notation.
\n\"Envision
\nAnswer:<\/p>\n

Question 19.
\nThe length of a bacterial cell is 5.2 \u00d7 10-6<\/sup> meter. Express the length of the cell in standard form.
\nAnswer:<\/p>\n

Question 20.
\nHigher Order Thinking<\/strong> Express the distance 4,300,000 meters using scientific notation in meters, and then in millimeters.
\nAnswer:<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 21.
\nWhich of the following numbers are written in scientific notation?
\n\"Envision 12 \u00d7 106<\/sup>
\n\"Envision 12
\n\"Envision 6.89 \u00d7 106<\/sup>
\n\"Envision 6.89
\n\"Envision 0.4
\n\"Envision 4 \u00d7 10-1<\/sup>
\nAnswer:<\/p>\n

Question 22.
\nJeana’s calculator display shows the number to the right.
\nPART A<\/strong>
\nExpress this number in scientific notation.
\n\"Envision<\/p>\n

PART B<\/strong><\/p>\n

Express this number in standard form.<\/p>\n

3-Act Mathematical Modeling: Hard-Working Organs<\/h3>\n

3-ACT MATH<\/strong><\/p>\n

\"Envision<\/p>\n

ACT 1<\/strong><\/p>\n

Question 1.
\nAfter watching the video, what is the first question that comes to mind?
\nAnswer:<\/p>\n

Question 2.
\nWrite the Main Question you will answer.
\nAnswer:<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Predict an answer to this Main Question. Explain your prediction.
\n\"Envision
\nAnswer:<\/p>\n

Question 4.
\nOn the number line below, write a number that is too small to be the answer. Write a number that is too large.
\n\"Envision
\nAnswer:<\/p>\n

Question 5.
\nPlot your prediction on the same number line.<\/p>\n

\u0410\u0421\u0422 2<\/strong><\/p>\n

Question 6.
\nWhat information in this situation would be helpful to know? How would you use that information?
\nAnswer:<\/p>\n

Question 7.
\nUse Appropriate Tools<\/strong> What tools can you use to solve the problem? Explain how you would use them strategically.
\n\"Envision
\nAnswer:<\/p>\n

Question 8.
\nModel with Math<\/strong> Represent the situation using mathematics. Use your representation to answer the Main Question.
\nAnswer:<\/p>\n

Question 9.
\nWhat is your answer to the Main Question? Is it greater or less than your prediction? Explain why.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3<\/strong><\/p>\n

Question 10.
\nWrite the answer you saw in the video.
\nAnswer;<\/p>\n

Question 11.
\nReasoning<\/strong> Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
\nAnswer:<\/p>\n

Question 12.
\nMake Sense and Persevere<\/strong> Would you change your model now that you know the answer? Explain.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3<\/strong><\/p>\n

Reflect<\/strong><\/p>\n

Question 13.
\nModel with Math<\/strong> Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
\nAnswer:<\/p>\n

Question 14.
\nGeneralize<\/strong> What pattern did you notice in your calculations? How did that pattern help you solve the problem?
\nAnswer:<\/p>\n

SEQUEL<\/strong><\/p>\n

Question 15.
\nUse Structure<\/strong> How many times does a heart beat in a lifetime? Use your solution to the Main Question to help you solve.
\n\"Envision
\nAnswer:<\/p>\n

Lesson 1.10 Operations with Numbers in Scientific Notation<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

The homecoming committee wants to fly an aerial banner over the football game. The banner is 1,280 inches long and 780 inches tall. How many different ways can the area of the banner be expressed?
\n\"Envision<\/p>\n

Focus on math practices<\/strong>
\nBe Precise Which of the solutions is easiest to manipulate?<\/p>\n

? Essential Question<\/strong>
\nHow does using scientific notation help when computing with very large or very small numbers?<\/p>\n

Try It!<\/strong><\/p>\n

The planet Venus is on average 2.5 \u00d7 107<\/sup> kilometers from Earth. The planet Mars is on average 2.25 \u00d7 108<\/sup> kilometers from Earth. When Venus, Earth, and Mars are aligned, what is the average distance from Venus to Mars?
\n\"Envision
\nAnswer:<\/p>\n

Convince Me!<\/strong>
\nIn Example 1 and the Try it, why did you move the decimal point to get the final answer?<\/p>\n

Try It!<\/strong><\/p>\n

There are 1 \u00d7 1014<\/sup> good bacteria in the human body. There are 2.6 x 1018<\/sup> good bacteria among the spectators in a soccer stadium. About how many spectators are in the stadium? Express your answer in scientific notation.<\/p>\n

KEY CONCEPT<\/strong>
\nOperations with very large or very small numbers can be carried out more efficiently using scientific notation. The properties of exponents apply when carrying out operations.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong>
\nHow does using scientific notation help when computing with very small or very large numbers?<\/p>\n

Question 2.
\nUse Structure<\/strong> When multiplying and dividing two numbers in scientific notation, why do you sometimes have to rewrite one factor?
\nAnswer:<\/p>\n

Question 3.
\nUse Structure<\/strong> For the sum of (5.2 \u00d7 104<\/sup>) and (6.95 \u00d7 104<\/sup>) in scientific notation, why will the power of 10 be 105<\/sup>?
\nAnswer:<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nA bacteriologist estimates that there are 5.2 \u00d7 104<\/sup> bacteria growing in each of 20 petri dishes. About how many bacteria in total are growing in the petri dishes? Express your answer in scientific notation.
\nAnswer:<\/p>\n

Question 5.
\nThe distance from Earth to the Moon is approximately 1.2 \u00d7 109<\/sup> feet. The Apollo 11 spacecraft was approximately 360 feet long. About how many spacecraft of that length would fit end to end from Earth to the Moon? Express your answer in scientific notation.
\n\"Envision
\nAnswer:<\/p>\n

Question 6.
\nThe mass of Mars is 6.42 \u00d7 1023<\/sup> kilograms. The mass of Mercury is 3.3 \u00d7 1023<\/sup> kilograms.
\na. What is the combined mass of Mars and Mercury expressed in scientific notation?
\nb. What is the difference in the mass of the two planets expressed in scientific notation?
\nAnswer:<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice<\/strong> In 7 and 8, perform the operation and express your answer in scientific notation.<\/p>\n

Question 7.
\n(7 \u00d7 10-6<\/sup>)(7 \u00d7 10-6<\/sup>)
\n\"Envision
\nAnswer:<\/p>\n

Question 8.
\n(3.76 \u00d7 105<\/sup>) + (7.44 \u00d7 105<\/sup>)
\n\"Envision
\nAnswer:<\/p>\n

Question 9.
\nWhat is the value of n in the equation
\n1.9 \u00d7 107<\/sup> = (1 \u00d7 105<\/sup>)(1.9 \u00d7 10n<\/sup>)?
\nAnswer:<\/p>\n

Question 10.
\nFind (5.3 \u00d7 103<\/sup>) – (8 \u00d7 102<\/sup>).
\nExpress your answer in scientific notation.
\nAnswer:<\/p>\n

Question 11.
\nWhat is the mass of 30,000 molecules? Express your answer in scientific notation.
\n\"Envision
\nAnswer:<\/p>\n

Question 12.
\nCritique Reasoning<\/strong> Your friend says that the product of 4.8 \u00d7 108<\/sup> and 2 \u00d7 10-3<\/sup> is 9.6 \u00d7 10-5<\/sup>. Is this answer correct? Explain.
\nAnswer:<\/p>\n

Question 13.
\nFind \\(\\frac{7.2 \\times 10^{-8}}{3 \\times 10^{-2}}\\). Write your answer in scientific white notation.
\nAnswer:<\/p>\n

Question 14.
\nA certain star is 4.3 \u00d7 102<\/sup> light years from Earth. One light year is about 5.9 \u00d7 1012<\/sup> miles. How far from Earth (in miles) is the star? Express your answer in scientific notation.<\/p>\n

Question 15.
\nThe total consumption of fruit juice in a particular country in 2006 was about 2.28 \u00d7 109<\/sup> gallons. The population of that country that year was 3 \u00d7 108<\/sup>. What was the average number of gallons consumed per person in the country in 2006?
\nAnswer:<\/p>\n

Question 16.
\nThe greatest distance between the Sun and Jupiter is about 8.166 \u00d7 108<\/sup> kilometers. The greatest distance between the Sun and Saturn is about 1.515 \u00d7 109<\/sup> kilometers. What is the difference between these two distances?
\nAnswer:<\/p>\n

Question 17.
\nWhat was the approximate number of pounds of garbage produced per person in the country in one year? Express your answer in scientific notation.
\n\"Envision
\nAnswer:<\/p>\n

Question 18.
\nHigher Order Thinking<\/strong>
\na. What is the value of n in the equation
\n1.5 \u00d7 1012<\/sup> = (5 \u00d7 105<\/sup>)(3 \u00d7 10n<\/sup>)?
\nb. Explain why the exponent on the left side of the equation is not equal to the sum of the exponents on the right side.<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 19.
\nFind (2.2 \u00d7 105<\/sup>) \u00f7 (4.4 \u00d7 10-3<\/sup>). When you regroup the factors, what do you notice about the quotient of the decimal factors? How does this affect the exponent of the quotient?
\nAnswer:<\/p>\n

Question 20.
\nWhich expression has the least value?
\nA. (4.7 \u00d7 104<\/sup>) + (8 \u00d7 104<\/sup>)
\nB. (7.08 \u00d7 103<\/sup>) + (2.21 \u00d7 103<\/sup>)
\nC. (5.43 \u00d7 108<\/sup>) \u2013 (2.33 \u00d7 108<\/sup>)
\nD. (9.35 \u00d7 106<\/sup>) \u2013 (6.7 \u00d7 106<\/sup>)
\nAnswer:<\/p>\n

Topic 1 Review<\/h3>\n

? Topic Essential Question<\/strong>
\nWhat are real numbers? How are real numbers used to solve problems?<\/p>\n

Vocabulary Review<\/strong><\/p>\n

\"Envision
\n\"Envision<\/p>\n

Use Vocabulary in Writing<\/strong>
\nUse vocabulary words to explain how to find the length of each side of a square garden with an area of 196 square inches.<\/p>\n

Concepts and Skills Review<\/strong><\/p>\n

LESSON 1.1 Rational Numbers as Decimals<\/strong><\/p>\n

Quick Review<\/strong><\/p>\n

You can write repeating decimals in fraction form by writing two equations. You multiply each side of one equation by a power of 10. Then you subtract the equations to eliminate the repeating decimal.<\/p>\n

Practice<\/strong>
\nWrite each number as a fraction or a mixed number.<\/p>\n

Question 1.
\n0.\\(\\overline{7}\\)
\nAnswer:<\/p>\n

Question 2.
\n0.0\\(\\overline{4}\\)
\nAnswer:<\/p>\n

Question 3.
\n4.\\(\\overline{45}\\)<\/p>\n

Question 4.
\n2.191919….
\nAnswer:<\/p>\n

LESSON 1.2 Understand Irrational Numbers<\/strong><\/p>\n

Quick Review<\/strong>
\nAn irrational number is a number that cannot be written in the form \\(\\frac{a}{b}\\), where a and b are integers and b \u2260 0. Rational and irrational numbers together make up the real number system.
\n\"Envision<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nDetermine which numbers are irrational. Select all that apply.
\n\"Envision \\(\\sqrt{36}\\)
\n\"Envision \\(\\sqrt{23}\\)
\n\"Envision -4.232323….
\n\"Envision 0.151551555….
\n\"Envision 0.3\\(\\overline{5}\\)
\n\"Envision \u03c0<\/p>\n

Question 2.
\nClassify -0.\\(\\overline{25}\\) as rational or irrational. Explain.
\nAnswer:<\/p>\n

LESSON 1.3 Compare and Order Real Numbers<\/strong><\/p>\n

Quick Review<\/strong>
\nTo compare and order real numbers, it helps to first write each number in decimal form.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nBetween which two whole numbers does \\(\\sqrt{89}\\) lie?
\n\\(\\sqrt{89}\\) is between \"Envision and \"Envision
\nAnswer:<\/p>\n

Question 2.
\nCompare and order the following numbers. Locate each number on a number line. 2.\\(\\overline{3}\\), \\(\\sqrt{8}\\), 2.5, 2\\(\\frac{1}{4}\\)
\nAnswer:<\/p>\n

LESSON 1.4 Evaluate Square Roots and Cube Roots<\/strong><\/p>\n

Quick Review<\/strong>
\nRemember that a perfect square is the square of an integer. A square root of a number is a number that when multiplied by itself is equal to the original number. Similarly, a perfect cube is the cube of an integer. A cube root of a number is a number that when cubed is equal to the original number.<\/p>\n

Practice<\/strong>
\nClassify each number as a perfect square, a perfect cube, both, or neither.<\/p>\n

Question 1.
\n27
\nAnswer:<\/p>\n

Question 2.
\n100
\nAnswer:<\/p>\n

Question 3.
\n64
\nAnswer:<\/p>\n

Question 4.
\n24
\nAnswer:<\/p>\n

Question 5.
\nA gift box is a cube with a volume of 512 cubic inches. What is the length of each edge of the box?
\nAnswer:<\/p>\n

LESSON 1.5 Solve Equations Using Square Roots and Cube Roots<\/strong><\/p>\n

Quick Review<\/strong><\/p>\n

You can use square roots to solve equations involving squares. You can use cube roots to solve equations involving cubes. Equations with square roots often have two solutions. Look at the context to see whether both solutions are valid.<\/p>\n

Practice Solve for x.<\/p>\n

Question 1.
\nx3<\/sup> = 64
\nAnswer:<\/p>\n

Question 2.
\nx2<\/sup> = 49
\nAnswer:<\/p>\n

Question 3.
\nx3<\/sup> = 25
\nAnswer:<\/p>\n

Question 4.
\nx2<\/sup> = 125
\nAnswer:<\/p>\n

Question 5.
\nA container has a cube shape. It has a volume of 216 cubic inches. What are the dimensions of one face of the container?<\/p>\n

LESSON 1.6 Use Properties of Integer Exponents<\/strong><\/p>\n

Quick Review<\/strong>
\nThese properties can help you write equivalent expressions that contain exponents.
\nProduct of Powers Property
\nam<\/sup>.an<\/sup> = am+n<\/sup>
\nPower of Powers Property
\n(am<\/sup>)n<\/sup> = amn<\/sup>
\nPower of Products Property
\nan<\/sup> \u2219 bn<\/sup> = (a \u2219 b)n<\/sup>
\nQuotient of Powers Property
\nam<\/sup> \u00f7 an<\/sup> = am-n<\/sup>, when a \u2260 0<\/p>\n

Practice<\/strong>
\nUse the properties of exponents to write an equivalent expression for each given expression.<\/p>\n

Question 1.
\n64<\/sup> \u2219 63<\/sup>
\nAnswer:<\/p>\n

Question 2.
\n(36<\/sup>)-2<\/sup>
\nAnswer:<\/p>\n

Question 3.
\n73<\/sup> \u2219 23<\/sup>
\nAnswer:<\/p>\n

Question 4.
\n410<\/sup> \u00f7 44<\/sup>
\nAnswer:<\/p>\n

LESSON 1.7 More Properties of Integer Exponents<\/strong><\/p>\n

Quick Review<\/strong>
\nThe Zero Exponent Property states that any nonzero number raised to the power of 0 is equal to 1. The Negative Exponent Property states that for any nonzero rational number a and integer n, a-n<\/sup> = \\(\\frac{1}{a^{n}}\\)<\/p>\n

Practice<\/strong>
\nWrite each expression using positive exponents.<\/p>\n

Question 1.
\n9-4<\/sup>
\nAnswer:<\/p>\n

Question 2.
\n\\(\\frac{1}{3^{-5}}\\)
\nAnswer:<\/p>\n

Evaluate each expression for x = 2 and y = 5<\/p>\n

Question 3.
\n-4x-2<\/sup> + 3y0<\/sup>
\nAnswer:<\/p>\n

Question 4.
\n2x0<\/sup>y-2<\/sup>
\nAnswer:<\/p>\n

LESSON 1.8 Use Powers of 10 to Estimate Quantities<\/strong><\/p>\n

Quick Review<\/strong>
\nYou can estimate very large and very small quantities by writing the number as a single digit times a power of 10.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nIn the year 2013 the population of California was about 38,332,521 people. Write the estimated population as a single digit times a power of 10.
\nAnswer:<\/p>\n

Question 2.
\nThe wavelength of green light is about 0.00000051 meter. What is this estimated wavelength as a single digit times a power of 10?
\nAnswer:<\/p>\n

Question 3.
\nThe land area of Connecticut is about 12,549,000,000 square meters. The land area of Rhode Island is about 2,707,000,000 square meters. How many times greater is the land area of Connecticut than the land area of Rhode Island?
\nAnswer:<\/p>\n

LESSON 1.9 Understand Scientific Notation<\/strong><\/p>\n

Quick Review<\/strong>
\nA number in scientific notation is written as a product of two factors, one greater than or equal to 1 and less than 10, and the other a power of 10.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nWrite 803,000,000 in scientific notation.
\nAnswer:<\/p>\n

Question 2.
\nWrite 0.0000000068 in scientific notation.
\nAnswer:<\/p>\n

Question 3.
\nWrite 1.359 \u00d7 105<\/sup> in standard form.
\nAnswer:<\/p>\n

Question 4.
\nThe radius of a hydrogen atom is 0.000000000025 meter. How would you express this radius in scientific notation?<\/p>\n

LESSON 1.10 Operations with Numbers in Scientific Notation<\/strong><\/p>\n

Quick Review<\/strong>
\nWhen multiplying and dividing numbers in scientific notation, multiply or divide the first factors. Then multiply or divide the powers of 10. When adding and subtracting numbers in scientific notation, first write the numbers with the same power of 10. Then add or subtract the first factors, and keep the same power of 10.
\nIf the decimal part of the result is not greater than or equal to 1 and less than 10, move the decimal point and adjust the exponent.<\/p>\n

Practice<\/strong>
\nPerform each operation. Express your answers in scientific notation.<\/p>\n

Question 1.
\n(2.8 \u00d7 104<\/sup>) \u00d7 (4 \u00d7 105<\/sup>)
\nAnswer:<\/p>\n

Question 2.
\n(6 \u00d7 109<\/sup>) \u00f7 (2.4 \u00d7 103<\/sup>)<\/p>\n

Question 3.
\n(4.1 \u00d7 104<\/sup>) + (5.6 \u00d7 106<\/sup>)
\nAnswer:<\/p>\n

Question 4.
\nThe population of Town A is 1.26 \u00d7 105<\/sup> people. The population of Town B is 2.8 \u00d7 104<\/sup> people. How many times greater is the population of Town A than the population of Town B?
\nAnswer:<\/p>\n

Topic 1 Fluency Practice<\/h3>\n

Crisscrossed<\/strong><\/p>\n

Solve each equation. Write your answers in the cross-number puzzle below. Each digit, negative sign, and decimal point of
\nyour answer goes in its own box.
\n\"Envision
\nAcross<\/strong>
\nA -377 = x \u2013 1,000
\nB x3<\/sup> = 1,000
\nC x3<\/sup> = -8
\nD x + 7 = -209
\nF x + 19 = -9
\nJ 14 + x = -9
\nL m – 2.02 = -0.58
\nM -3.09 + x = -0.7
\nN -2.49 = -5 + x
\nQ x – 3.5 = -3.1
\nT q \u2013 0.63 = 1.16
\nV 8.3 + x = 12.1<\/p>\n

Down<\/strong>
\nA y – 11 = 49
\nB x + 8 = 20
\nC z3<\/sup> = -1,331
\nD 11 + x = 3
\nE x – 14 -7.96
\nF 14 + x = -19
\nG d + 200 = 95
\nH x2<\/sup> = 144
\nK -12 = t – 15.95
\nP 0.3 + x = 11
\nR x \u2013 3 = -21
\nS – 7 = -70 + y<\/p>\n","protected":false},"excerpt":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 8 Answer Key\u00a0Topic 1 Real Numbers\u00a0regularly and improve your accuracy in solving questions. enVision Math Common Core 8th Grade Answers Key Topic 1 Real Numbers enVision STEM Project Did You Know? Natural resources are materials that occur in nature, such as water, fossil fuels, wood, and minerals. Natural …<\/p>\n","protected":false},"author":1,"featured_media":25554,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/envisionmathanswerkey.com\/wp-content\/uploads\/2021\/09\/Envision-Math-Common-Core-Grade-8-Answer-Key-Topic-1-Real-Numbers.png","_links":{"self":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/4668"}],"collection":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/comments?post=4668"}],"version-history":[{"count":3,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/4668\/revisions"}],"predecessor-version":[{"id":27290,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/4668\/revisions\/27290"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/media\/25554"}],"wp:attachment":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/media?parent=4668"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/categories?post=4668"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/tags?post=4668"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}