Powers and Exponents1-4

Warm UpWarm Up

Lesson PresentationLesson Presentation

California StandardsCalifornia Standards

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Powers and Exponents1-4

Warm UpSimplify.

1. 2(2)

2. (–2)(–2)

3. (–2)(–2)(–2)

4. 3(3)(3)

4

4

–8

27

495.

Powers and Exponents1-4

Preparation for 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.

California Standards

Powers and Exponents1-4

powerbaseexponent

Vocabulary

Powers and Exponents1-4

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.

The base, 3, is the number that is used as a factor.

32 The exponent, 2, tellshow many times thebase, 3, is used as afactor.

Powers and Exponents1-4When 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.

s

s

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.

s s

s

Powers and Exponents1-4

Write the power represented by the geometric model.

55

5 The figure is 5 units long, 5 units wide, and 5 units tall. 5 5 5

The factor 5 is used 3 times.53

Additional Example 1A: Writing Powers for Geometric Models

Powers and Exponents1-4

Write the power represented by the geometric model.

The figure is 6 units long and 6 units wide. 6 x 6

The factor 6 is used 2 times.62

6

6

Additional Example 1B: Writing Powers for

Geometric Models

Powers and Exponents1-4

Write the power represented by each geometric model.

a.

22

b.

Check It Out! Example 1

x

xx

The figure is 2 units long and 2 units wide. 2 2

The factor 2 is used 2 times.

The figure is x units long, x units wide, and x units tall. x x x

The factor x is used 3 times.x3

Powers and Exponents1-4There are no easy geometric models for numbers raised to exponents greater than 3, but you can still write them using repeated multiplication or with a base and exponent.

3 to the second power, or 3 squared

3 3 3 3 3

Multiplication Power ValueWords

3 3 3 3

3 3 3

3 3

33 to the first power

3 to the third power, or 3 cubed

3 to the fourth power

3 to the fifth power

3

9

27

81

243

31

32

33

34

35

Reading Exponents

Powers and Exponents1-4

Caution!In the expression –5², 5 is the base because

the negative sign is not in parentheses. In the expression (–2)³, –2 is the base because of the parentheses.

Powers and Exponents1-4

Simplify each expression.

A. (–6)3

(–6)(–6)(–6)

–216

B. –102

–1 • 10 • 10

–100

Use –6 as a factor 3 times.

Find the product of –1 andtwo 10’s.

Additional Example 2: Evaluating Powers

Think of a negative sign in front of a power as multiplying by a –1.

Powers and Exponents1-4

Use as a factor 2 times.2 9

Simplify the expression.

C.

29

29

Additional Example 2: Evaluating Powers

= 4

81

29

29

Powers and Exponents1-4

Simplify each expression.

a. (–5)3

b. –62

Check It Out! Example 2

(–5)(–5)(–5) Use –5 as a factor 3 times.

–125

–1 6 6

–36

Think of a negative sign in front of a power as multiplying by –1.

Find the product of –1 andtwo 6’s.

Powers and Exponents1-4

Check It Out! Example 2

Simplify the expression.

c.

Use as a factor 3 times.34

2764

Powers and Exponents1-4

Write each number as a power of the given base.

A. 64; base 8

8 8

82

B. 81; base –3

(–3)(–3)(–3)(–3)

(–3)4

The product of two 8’s is 64.

The product of four –3’s is 81.

Additional Example 3: Writing Powers

Powers and Exponents1-4

Write each number as a power of a given base.

a. 64; base 8

b. –27; base –3

Check It Out! Example 3

8 8 The product of two 8’s is 64.

8²

(–3)(–3)(–3)

(–3)3

The product of three –3’s is –27.

Powers and Exponents1-4

In case of a school closing, the PTApresident calls 3 families. Each of these families calls 3 other families,and so on. How many families will have been called in the 4th round of calls?

The answer will be the number of familiescontacted in the 4th round of calls.

Additional Example 4: Problem-Solving Application

Understand the problem11

List the important information:

• The PTA president calls 3 families.• Each family then calls 3 more families.

Powers and Exponents1-4

Draw a diagram to show the number of families called in each round of calls.

22 Make a Plan

Additional Example 4 Continued

2nd round of calls

1st round of calls

PTA President

Powers and Exponents1-4

Notice that after each round of calls the number of families contacted is a power of 3.

1st round of calls: 1 3 = 3 or 31 families contacted

So, in the 4th round of calls, 34 families will havebeen contacted.

34 = 3 3 3 3 = 81

Multiply four 3’s.

In the fourth round of calls, 81 families will have been contacted.

2nd round of calls: 3 3 = 9 or 32 families contacted

3rd round of calls: 9 3 = 27 or 33 families contacted

Solve33

Additional Example 4 Continued

Powers and Exponents1-4

Drawing a diagram helps you visualize the problem, but the numbers become toolarge for a diagram. The diagram helps you recognize the pattern of multiplying by 3 so that you can write the number as apower of 3.

Look Back44

Additional Example 4 Continued

Powers and Exponents1-4

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

Check It Out! Example 4

28After each hour, the

number of bacteria is a power of 2.

2 2 2 2 2 2 2 2 Multiply eight 2’s.

The product of eight 2’s.256

Powers and Exponents1-4

1. Write the power represented by the geometric model.

n

n n2

Simplify each expression.

2.

4. 6

3. –63

5. (–2)6

−216

216 64

Lesson Quiz

Write each number as a power of the given base.

6. 343; base 7 7. 10,000; base 1073 104