Exponent 3

Chapter Chapter 55Section Section 11

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


The Product Rule and Power Rules for Exponents

Use exponents.Use the product rule for exponents.Use the rule (am)n = amn.Use the rule (ab)m = ambn.Use the ruleUse combinations of rules.Use the rules for exponents in a geometric application.

11

44

33

22

66

55

5.5.115.5.11

77

.m m

m

a a

b b


Objective 11

Slide 5.1 - 3

Use exponents.


Use exponents.

Recall from Section 1.2 that in the expression 52, the number 5 is the base and 2 is the exponent or power. The expression 52 is called an exponential expression. Although we do not usually write the exponent when it is 1, in general, for any quantity a, a1 = a.

Slide 5.1 - 4


EXAMPLE 1

Write 2 · 2 · 2 in exponential form and evaluate.

Solution:

Using Exponents

Slide 5.1 - 5

2 2 2 832


Evaluate. Name the base and the exponent.

EXAMPLE 2 Evaluating Exponential Expressions

Slide 5.1 - 6

62Solution:

64

62 64

Base: Exponent:2 6

Base Exponent2 6

1 2 2 2 2 2 2

2 2 2 2 2 2

Note the difference between these two examples. The absence of parentheses in the first part indicate that the exponent applies only to the base 2, not −2.


Objective 22

Slide 5.1 - 7

Use the product rule for exponents.


Use the product rule for exponents.By the definition of exponents,

Generalizing from this example

suggests the product rule for exponents.

Slide 5.1 - 8

For any positive integers m and n, a m · a n = a m + n.

(Keep the same base; add the exponents.)

Example: 62 · 65 = 67

4 32 2 2 2 2 2 2 2 2

4 3 4 3 72 2 2 2

2 2 2 2 2 2 2 72

Do not multiply the bases when using the product rule. Keep the same base and add the exponents. For example

62 · 65 = 67, not 367.


EXAMPLE 3

Solution:

Using the Product Rule

Slide 5.1 - 9

5 37 7

5 84 3 p 5 3

7 8

7 1312 p

Use the product rule for exponents to find each product if possible.

a)

b)

c)

d)

e)

f)

5 84 3p p4m m

2 5 6z z z2 54 3

4 26 6

2 5 6z

1 4m 5m13z

13323888 The product rule does not apply.

The product rule does not apply.Be sure you understand the difference between adding and multiplying exponential expressions. For example,

but 3 3 3 38 5 8 5 3 ,1x x x x

3 3 3 3 68 5 8 5 4 .0x x x x


Objective 33

Slide 5.1 - 10

Use the rule (am)n = amn.


We can simplify an expression such as (83)2 with the product rule for exponents.

Use the rule (am)n = amn.

Slide 5.1 - 11

23 3 3 3 3 68 8 8 8 8 The

exponents in (83)2 are multiplied to give the exponent in 86.

This example suggests power rule (a) for exponents.

For any positive number integers m and n, (am)n = amn.(Raise a power to a power by multiplying exponents.)Example: 42 2 4 83 3 3


EXAMPLE 4

Solution:

1062 56

20z

526

4 5z

Using Power Rule (a)

Slide 5.1 - 12

Simplify.

54z

Be careful not to confuse the product rule, where 42 · 43 = 42+3 = 45 =1024

with the power rule (a) where (42)3 = 42 · 3 = 46 = 4096.


Objective 44

Use the rule (ab)m = ambm.

Slide 5.1 - 13


Use the rule (ab)m = ambm.

We can rewrite the expression (4x)3 as follows.

Slide 5.1 - 14

34 4 4 4x x x x

4 4 4 x x x 3 34 x

This example suggests power rule (b) for exponents.

For any positive integer m, (ab)m = ambm. (Raise a product to a power by raising each factor to the power.)

Example: 5 5 52 2p p


EXAMPLE 5

Simplify.Solution:

52 43a b

Using Power Rule (b)

Slide 5.1 - 15

323m

5 55 2 43 a b

33 21 3 m

10 20243a b

627m

Power rule (b) does not apply to a sum. For example,

, but 2 2 24 4x x 2 2 24 4 .x x

Use power rule (b) only if there is one term inside parentheses.


Objective 55

Use the rule

Slide 5.1 - 16

m m

m

a a

b b

.


Use the rule

Slide 5.1 - 17

Since the quotient can be written as we use this fact

and power rule (b) to get power rule (c) for exponents.

For any positive integer m,

(Raise a quotient to a power by raising both numerator and denominator to the power.)

Example:

m m

m

a a

b b

.

a

b1

,ab

m m

m

a ab

b b

0 .

2 2

2

5 5

3 3


EXAMPLE 6

Simplify.Solution:

3

30x

x

Using Power Rule (c)

Slide 5.1 - 18

51

3

3

3

3

x

5

5

1

3

3

27

x

1

243

In general, 1n = 1, for any integer n.


The rules for exponents discussed in this section are summarized in the box.

Slide 5.1 - 19

Rules of Exponents

These rules are basic to the study of algebra and should be memorized.


Objective 66

Use combinations of rules.

Slide 5.1 - 20


EXAMPLE 7

4

212

5x

Slide 5.1 - 21

Simplify

22 3

2

5

3

k

625

9

k

3 43 3 3 42 21 3 x y x y

2 24

4

21

5 1

x

24

625

x

3 6 8 41 27 x y x y 11 1027x y

Solution:

Use Combinations of Rules

235

3

k

3 42 23xy x y


Objective 77

Use the rules for exponents in a geometric application.

Slide 5.1 - 22


EXAMPLE 8 Using Area Formulas

Slide 5.1 - 23

Find an expression that represents the area of the figure.

A LW

2 44 8A x x2 44 8A x

Solution:

632A x 632A x

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