### Key Concepts

After this lesson, students will be able to:

- Understand and Represent Exponents.
- Identify the base and exponent in a power.
- Write powers for repeated multiplication
- Write powers in expand and word form
- Evaluate Exponents
- Evaluate Expressions with Exponents.

## Essential Question

How can you write and evaluate numbers with exponents?

**EXAMPLE 1: **

2 x 2 x 2 represents the number of cells after 1 hour if there is 1 cell at the start.

How can you write this expression using exponents? How many cells will there be after 1 hour?

**Reasoning **

Repeated multiplication can be represented in more than one way.

You can use an exponent to write a **repeated multiplication** of a number.

A number that can be written using exponents is called a power.

You can use **repeated multiplication** to evaluate, or find the value of a power.

#### Complete the table below:

**Try It! **

There are 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 bacteria cells after 3 hours. Write the repeated multiplication as a power, and then evaluate.

**Convince Me! **

Why can you represent the number of cells after two hours as the power 2⁶ ?

Yes, First hour = 2 x 2 x 2 = 2³ = 8

Second hour = 2 times 2 x 2 x 2= (2 x 2 x 2) x (2 x 2 x 2) = 2⁶ = 64

Third hour = 3 times 2 x 2 x 2 = (2 x 2 x 2) x (2 x 2 x 2 ) x ( 2 x 2 x 2 )= 2⁹ =512

**Evaluate Exponents **

**EXAMPLE 2:**

**How can you evaluate 2°?**

The base is 2. The exponent is 0. Make a table and look for a pattern.

Each value equals the previous value multiplied by 2.

1 x 2 = 2, so the value of 2° is 1.

Generalize,

Any non-zero number raised to an exponent of zero has a value of 1.

**How can you evaluate**1.2⁴ ?

Solution:

The base is 1.2. The exponent is 4.

1.2⁴ = 1.2 x 1.2 x 1.2 x 1.2

= 1.44 x 1.2 x 1.2 (∵Multiply by the first two factors)

= 1.728 x 1.2 (∵Multiply by the third factor)

∴1.2⁴ =2.0736 (∵Multiply by the fourth factor)

1.2⁴ = 2.0736

**EXAMPLE 3:**

Julia calculated the foil as 1.9 x 10⁵units thick. Thom calculated the foil as 183,000 units thick. Which calculation represents the greater thickness for the foil?

Evaluate the expression: 1.9 x10⁵

Solution:

10⁵= 10 x 10 x 10 x 10 x 10 = 100000

Multiply by the decimal: 1.9 X 100,000 = 190,000

Compare the numbers.

190,000 > 183,000 Julia’s calculation represents the greater thickness for the foil.

Try It!

Rafael calculated the foil as 1.8 x10⁵ units thick. Evaluate Rafael’s expression.

**Solution: **

1.8 x10⁵ =1.8 x 100000 =180,000.

**Practice & Problem Solving:**

**Write the exponent for each expression**

- A company rents two storage units. Both units are cube-shaped. What is the difference in the volume of the two storage units? Note that the volume of a cube is s3, where s is the side length. Explain.

**Solution: **

Given that,

Two storage units which are cube shaped with side lengths of 8 ft and 6.5 ft, respectively.

Volume of the first storage unit = s³

= (8)³

=8 x 8 x 8

=512 square fts

Volume of the second storage unit = s³

= (6.5)³

=6.5 x 6.5 x 6.5

=274.625 square fts

Difference of two storage units in volume

= 512 square fts – 274.625 square fts

=237.375 square fts.

∴Difference of two storage units in volume = 237.375 square fts.

- Malik read that the land area of Alaska is about 5.7 x 10⁵ square miles. About how many square miles is the land area of Alaska?

**Solution: **

Given that,

Land area of Alaska = 5.7 x 10⁵

square miles

5.7 x 10⁵

= 5.7 x 10 x 10 x 10 x 10 x 10

= 5.7 x 100,000

=570,000

∴The land area of Alaska = 570,000 square miles

### Higher Order Thinking

- Zach invested $50 and tripled his money in two years. Kayla also invested $50, and after two years the amount was equal to 50 to the third power. Who had more money after two years? Explain.

**Solution: **

Given that,

Money invested by Zach = $50

Zach tripled his money in two years

Then,

Zach money after two years = 3 x 50

=150

Zach money after two years= $150

Money invested by Kayla = $50

Her amount was equal to 50 to the third power

Then,

Kayla money after two years = 50³

=50 x 50 x 50

= $125,000

∴Kayla has more money after two years

### Check your knowledge:

**Answer the following**

- Write 81 as the repeated multiplication of 3s. Then write it as power.

- Write 125 as the repeated multiplication of 5s. Then write it as a power.

- What is 0.75 x 0.75 x 0.75 x 0.75 written as power?

What is 3/8 x 3/8 x 3/8 written as power?

**Evaluate each power.**

- (1/6)²

- 45⁰

- 0.1⁵

- 7³

**Evaluate each expression.**

- 4.5 x 10³

- 0.6 x 10⁶

- 3.4 x 10⁰

**Answers:**

- 81 = 3 x 27

=3 x 3 x 9

=3 x 3 x 3 x 3

= 3⁴

∴ 81 = 3⁴

- 125 = 5 x 25

= 5 x 5 x 5

= 5³

∴ 125 = 5³

- 0.75 x 0.75 x 0.75 x 0.75 = 0.75⁴

- 3/8 x 3/8 x 3/8 = (3/8)³
- (1/6)² = 1/6 x 1/6

1/36

- 45⁰ = 1

- 0.1⁵= 0.1 x 0.1 x 0.1 x 0.1 x 0.1

=0.00001

- 7³= 7 x 7 x 7

= 343

- 4.5 x 10³= 4.5 x 10 x 10 x 10

=4.5 x 1000

=4500

- 0.6 x 10⁶ = 0.6 x 10 x 10 x 10 x 10 x 10 x 10

=0.6 x 1,000,000

=600,000

- 3.4 x 10⁰ = 3.4 x 1

=3

**Key concept covered**

- Understand and represent exponents.
- Identify the base and the exponent in a power.
- Write powers for repeated multiplication.
- Write powers in expanded and word form.
- Evaluate exponents.
- Evaluate expressions with exponents.

### Concept map

**3.1 Understand and Represent Exponents**

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