CONFIDENTIAL 1

Powers andPowers andExponentsExponents

CONFIDENTIAL 2

Warm UpWarm Up

1)

2)

3)

4)

Multiply or divide if possible:

CONFIDENTIAL 3

Powers and ExponentsPowers and Exponents

When bacteria divide, their number increasesexponentially. This means that the number ofbacteria is multiplied by the same factor each

time the bacteria divide. Instead of writingrepeated multiplication to express a product,

you can use a power.

A power is an expression written with an exponent and a base or the value of such an

expression. 32 is an example of a power.

CONFIDENTIAL 4

When a number is raised to the second power, we usually say it is “squared.” The area of a

square is s · s = s2 , where s is the side length.

When a number is raised to the third power, we usually say it is “cubed.” The volume of a cube is

s · s · s = s3 , where s is the side length.

CONFIDENTIAL 5

Writing Powers for Geometric ModelsWriting Powers for Geometric Models

There are 3 rows of 3 dots. 3 × 3The factor 3 is used 2 times.

Write the power represented by each geometric model.

The figure is 4 cubes long, 4 cubes wide, and 4 cubes tall. 4 × 4 × 4 The factor 4 is used 3 times.

CONFIDENTIAL 6

Now you try!

Write the power represented by each geometric model.

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There are no easy geometric models for numbers raised to exponents greater than 3, but you can still write them

using repeated multiplication or a base and exponent.

CONFIDENTIAL 8

Evaluating PowersEvaluating PowersSimplify each expression.

A) (-2)3

=(-2) (-2) (-2) = -8

B) -52

= -1 · 5 · 5 = -1 · 25 = -25

Think of a negative sign in front of a power as multiplying by -1. Find the product of -1 and two 5’s.

Use -2 as a factor 3 times.

C) (2)2

(3)2

= (2) . (2) (3) (3) = 4 9

Use 2 as a factor 2 times. 3

CONFIDENTIAL 9

Now you try!

Simplify each expression.

2a) ( -5)3

2b) – 62

2c) (3)2

(4)2

CONFIDENTIAL 10

Writing PowersWriting Powers

The product of three 2’s is 8.

Write each number as a power of the given base.

The product of three -5’s is -125.

A) 8; base 2 = 2 · 2 · 2 = 23

B) -125; base -5 = (-5) (-5) (-5) = (-5)3

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Now you try!

Write each number as a power of the given base.

3a) 64; base 8

3b) -27; base -3

CONFIDENTIAL 12

Problem-Solving ApplicationProblem-Solving Application

A certain bacterium splits into 2 bacteria every hour. There is 1 bacterium on a slide. How many bacteria will

be on the slide after 6 hours?

• There is 1 bacterium on a slide that divides into 2 bacteria.• Each bacterium then divides into 2 more bacteria.

The diagram shows the number of bacteria after each hour.

Notice that after each hour, the number of bacteria is a power of 2.

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After 1 hour: 1 · 2 = 2 or 21 bacteria on the slide

After 2 hours: 2 · 2 = 4 or 22 bacteria on the slide

After 3 hours: 4 · 2 = 8 or 23 bacteria on the slide

So, after the 6th hour, there will be 2 6 bacteria.

26 = 2 · 2 · 2 · 2 · 2 · 2 = 64

After 6 hours, there will be 64 bacteria on the slide.

Multiply six 2’s.

The numbers become too large for a diagram quickly, but a diagram helps you recognize a pattern. Then you can write

the numbers as powers of 2.

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Now you try!

4) A certain bacterium splits into 2 bacteria every hour. There is 1 bacterium on a slide.

How many bacteria will be on the slide after 8 hours?

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Assessment

Write the power represented by each geometric model.

1) 2)

3)

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Simplify each expression:

4) 72

5) (-2)4

6) (-2)5

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Write each number as a power of the given base.

7) 81; base 9

8) 100,000; base 10

9) -64; base -4

CONFIDENTIAL 18

10) Jan wants to predict the number of hits she will get on her Web page. Her Web

page received 3 hits during the first week it was posted. If the number of hits triples every week, how many hits will the Web

page receive during the 5th week?

CONFIDENTIAL 19

Powers and ExponentsPowers and Exponents

When bacteria divide, their number increasesexponentially. This means that the number ofbacteria is multiplied by the same factor each

time the bacteria divide. Instead of writingrepeated multiplication to express a product,

you can use a power.

A power is an expression written with an exponent and a base or the value of such an

expression. 32 is an example of a power.

Let’s review

CONFIDENTIAL 20

When a number is raised to the second power, we usually say it is “squared.” The area of a

square is s · s = s2 , where s is the side length.

When a number is raised to the third power, we usually say it is “cubed.” The volume of a cube is

s · s · s = s3 , where s is the side length.

CONFIDENTIAL 21

Writing Powers for Geometric ModelsWriting Powers for Geometric Models

There are 3 rows of 3 dots. 3 × 3The factor 3 is used 2 times.

Write the power represented by each geometric model.

The figure is 4 cubes long, 4 cubes wide, and 4 cubes tall. 4 × 4 × 4 The factor 4 is used 3 times.

CONFIDENTIAL 22

Evaluating PowersEvaluating PowersSimplify each expression.

A) (-2)3

=(-2) (-2) (-2) = -8

B) -52

= -1 · 5 · 5 = -1 · 25 = -25

Think of a negative sign in front of a power as multiplying by -1. Find the product of -1 and two 5’s.

Use -2 as a factor 3 times.

C) (2)2

(3)2

= (2) . (2) (3) (3) = 4 9

Use 2 as a factor 2 times. 3

CONFIDENTIAL 23

Problem-Solving ApplicationProblem-Solving Application

A certain bacterium splits into 2 bacteria every hour. There is 1 bacterium on a slide. How many bacteria will

be on the slide after 6 hours?

• There is 1 bacterium on a slide that divides into 2 bacteria.• Each bacterium then divides into 2 more bacteria.

The diagram shows the number of bacteria after each hour.

Notice that after each hour, the number of bacteria is a power of 2.

CONFIDENTIAL 24

After 1 hour: 1 · 2 = 2 or 21 bacteria on the slide

After 2 hours: 2 · 2 = 4 or 22 bacteria on the slide

After 3 hours: 4 · 2 = 8 or 23 bacteria on the slide

So, after the 6th hour, there will be 2 6 bacteria.

26 = 2 · 2 · 2 · 2 · 2 · 2 = 64

After 6 hours, there will be 64 bacteria on the slide.

Multiply six 2’s.

The numbers become too large for a diagram quickly, but a diagram helps you recognize a pattern. Then you can write

the numbers as powers of 2.

CONFIDENTIAL 25

You did a great job You did a great job today!today!