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# Understand and Represent Exponents

### 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.

#### 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:

1. 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.

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

1. 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.

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

1. 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

1. Write 81 as the repeated multiplication of 3s. Then write it as power.
1. Write 125 as the repeated multiplication of 5s. Then write it as a power.
1. 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. (1/6)²
1. 45⁰
1. 0.1⁵
1. 7³

Evaluate each expression.

1. 4.5 x 10³
1. 0.6 x 10⁶
1. 3.4 x 10⁰

1. 81 = 3 x 27

=3 x 3 x 9

=3 x 3 x 3 x 3

= 3⁴

∴ 81 = 3⁴

1. 125 = 5 x 25

= 5 x 5 x 5

= 5³

∴ 125 = 5³

1. 0.75 x 0.75 x 0.75 x 0.75 = 0.75⁴
1. 3/8 x 3/8 x 3/8 = (3/8)³
2. (1/6)² = 1/6 x 1/6
1/36
1. 45⁰ = 1
1. 0.1⁵= 0.1 x 0.1 x 0.1 x 0.1 x 0.1

=0.00001

1. 7³= 7 x 7 x 7
= 343
1. 4.5 x 10³= 4.5 x 10 x 10 x 10
=4.5 x 1000
=4500
1. 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
1. 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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