## Envision Math 6th Grade Textbook Answer Key Topic 1.3 Exponents and Place Value

**Exponents and Place Value**

How can you write a number

using exponents?

Answer:

Each place in a place-value chart has a value that is 10 times as great as the place to its right. Use this pattern to write 1,000,000 as repeated multiplication.

**Another Example**

How do you write the expanded form of a number using exponents?

Answer:

Standard form: 562,384

Expanded form: (5 × 100,000) + (6 × 10,000) + (2 × 1,000) + (3 × 100) + (8 × 10) + 4

Expanded form

using exponents: (5 × 10^{5}) + (6 × 10^{4}) + (2 × 10^{3}) + (3 × 10^{2}) + (8 × 10^{1} ) + (4 × 10^{0})

Any number raised to the first power always equals that number. 10^{1} = 10.

**Explain It**

Question 1.

How many times is 9 used as a factor in the exponent 98?

Answer:

8 times

Explanation:

Question 2.

Why does 3 × 100 = 3?

Answer:

10^{0} = 1, so 3 × 10^{0} = 3

**Other Examples**

Write each in exponential form.

100,000 = 10^{5} 10 × 10 × 10 = 10^{3} 1 trillion = 10^{12}

Evaluate numbers in exponential form.

5^{3} = 5 × 5 × 5 = 125 3^{4} = 3 × 3 × 3 × 3 = 81

You can write the repeated multiplication of a number in exponential form.

Each place in the place-value chart can be written using an exponent.

**Guided Practice**

**Do you know HOW?**

Question 1.

Write 10,000 as repeated multiplication.

Answer:

10,000 = 10 × 10 × 10 × 10

Explanation:

Question 2.

Write 7 × 7 × 7 × 7 in exponential form.

Answer:

7^{4}

Explanation:

Question 3.

Write 37,169 in expanded form using exponents.

Answer:

(3 × 10^{4}) + (7 × 10^{3}) + (1 × 10^{2}) + (6 × 10^{1}) + (9 × 10^{0})

Explanation:

Question 4.

Write 5^{3} in standard form.

Answer:

125

Explanation:

**Do you UNDERSTAND?**

Question 5.

In the example at the top, why was the number 10 used as the base to write 1,000,000 in exponential form?

Answer:

See margin.

Explanation:

Question 6.

Using the example, how many times would 10 be repeatedly multiplied to equal 100,000?

Answer:

5 times. 100,000 = 10 × 10 × 10 × 10 × 10 = 10

Explanation:

Question 7.

How many zeros are in 107 when it is written in standard form?

Answer:

7

Explanation:

**Independent Practice**

Leveled Practice What number is the base?

Question 8.

4^{9}

Answer:

4

Explanation:

Question 9.

17^{9}

Answer:

17

Explanation:

What number is the exponent?

Question 10.

31^{9}

Answer:

9

Explanation:

Question 11.

2^{100}

Answer:

100

Explanation:

Write each in exponential form.

Question 12.

1,000

Answer:

10^{3}

Explanation:

Question 13.

1,000,000,000

Answer:

10^{9}

Explanation:

Question 14.

10 × 10 × 10 × 10 × 10

Answer:

10^{5}

Explanation:

Write each number in expanded form using exponents.

Answer:

See margin

Explanation:

Question 15.

841

Answer:

Question 16.

5,832

Answer:

Question 17.

1,874,161

Answer:

Question 18.

22,600,000

Answer:

Evaluate 19 through 22.

Question 19.

6^{2} = ☐

Answer:

36

Explanation:

Question 20.

10^{4} = ☐

Answer:

10,000

Explanation:

Question 21.

4^{3} = ☐

Answer:

64

Explanation:

Question 22.

2^{7} = ☐

Answer:

128

Explanation:

**Problem Solving**

Question 23.

The population of one U.S. state is approximately 33,871,648. What is this number in expanded form using exponents?

Answer:

See margin

Explanation:

Question 24.

Reasoning What number raised to both the first power and the second power equals 1?

Answer:

1; 1^{1} = 1 and 1^{2} = 1

Explanation:

Question 25.

Writing to Explain Explain how to compare 24 and 42.

Answer:

2^{4} = 2 × 2 × 2 × 2 = 16; 4^{2} = 4 × 4 = 16; So 2^{4} = 4^{2}.

Explanation:

Question 26.

In Exercise 23, what is the place of the digit 7?

A. hundreds

B. thousands

C. ten thousands

D. millions

Answer:

C. ten thousands

Explanation:

Question 27.

Writing to Explain Kalesha was asked to write 80,808 in expanded form using exponents. Her response was (8 × 10^{2}) + (8 × 10^{1}) + (8 × 10^{0}). Explain where she made mistakes and write the correct response.

Answer:

See margin

Explanation:

Question 28.

Think About the Process You invest $1 in a mutual fund. Every 8 years, your money doubles. If you don’t add more money, which expression shows how much your investment is worth after 48 years?

A. 1 48

B. 1 × 2 × 2 × 2 × 2 × 2

C. 1 + 2 + 2 + 2 + 2 + 2 + 2

D. 1 × 2 × 2 × 2 × 2 × 2 × 2

Answer:

Explanation:

D. 1 × 2 × 2 × 2 × 2 × 2 × 2

Question 29.

Number Sense Using the map, write the population of the United States in expanded form using exponents.

Answer:

See margin

Explanation:

Question 30.

In 1900, there were 76,803,887 people in the United States. How many more people were there in the United States in a recent year than in 1900?

Answer:

See margin

Explanation:

**Algebra Connections**

Solution Pairs

An equation is a mathematical sentence that uses an equals sign to show that two expressions are equal. Any values that make an equation true are solutions to the equation.

An inequality is a mathematical sentence that contains <, >, ≤, or ≥. Any value that makes the inequality true is a solution. You can graph the solutions of an inequality on a number line.

**Example:** Find two values for each variable that make the equation,

y = x + 3, true.

If x = 1, then y = 1 + 3 = 4 is true.

If x = 5, then y = 5 + 3 = 8 is true.

(1, 4) and (5, 8) are solution pairs.

**Example:** Graph three values that make the inequality, x > 3, true.

x = 3.1, x = 4, x = 5

Draw a number line. Plot three points that are greater than 3

For 1 through 4, copy the table and find two values for each variable that make the equation true.

Question 1.

y = 4 + x

Answer:

Explanation:

Question 2.

b = a – 2

Answer:

Explanation:

Question 3.

t = 3w

Answer:

Explanation:

Question 4.

y = x ÷ 2

Answer:

Explanation:

Question 5.

Copy the number line and graph 3 values that make the inequality, d ≥ 9, true.

Answer:

Any 3 points right of 9; or 9

Explanation:

Question 6.

Copy the number line and graph 3 values that make the inequality,\(\frac{x}{3}\) < 4, true.

Answer:

Any 3 points to left of 12

Explanation: