4.1 The Product Rule and Power Rules for Exponents

Objective 1

Use exponents.

Slide 4.1-3

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 4.1-4

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

Solution:

2 2 2 832

Slide 4.1-5

Using ExponentsCLASSROOM EXAMPLE 1

Evaluate. Name the base and the exponent.

Solution:

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 indicates that the exponent applies only to the base 2, not −2.

Slide 4.1-6

Evaluating Exponential Expressions

62

CLASSROOM EXAMPLE 2

Objective 2

Use the product rule for exponents.

Slide 4.1-7

Use the product rule for exponents.

By the definition of exponents,

Product Rule for ExponentsFor 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 2 2 2 2 2 2 2

Generalizing from this example

suggests the product rule for exponents.

4 3 4 3 72 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.

Slide 4.1-8

Solution:

5 37 7

5 84 3 p

5 37

87

1312p

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

5 84 3p p

4m m2 5 6z z z

2 54 34 26 6

2 5 6z

1 4m 5m

13z

1332

3888 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,

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

Slide 4.1-9

Using the Product RuleCLASSROOM EXAMPLE 3

87

Objective 3

Use the rule (am)n = amn.

Slide 4.1-10

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

23 3 3 3 3 68 8 8 8 8

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

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

Slide 4.1-11

Use the rule (am)n = amn.

Solution:

1062 56

20z

526

4 5z

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.

Slide 4.1-12

Using Power Rule (a)CLASSROOM EXAMPLE 4

Objective 4

Use the rule (ab)m = am bm.

Slide 4.1-13

Use the rule (ab)m = ambm.

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

34 4 4 4x x x x

4 4 4 x x x 3 34 x

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

Slide 4.1-14

Simplify.

Solution:

52 43a b

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.

Slide 4.1-15

Using Power Rule (b)CLASSROOM EXAMPLE 5

Objective 5 m m

m

a ab b

Use the rule .

Slide 4.1-16

Use the rule

Since the quotient can be written as we use this fact and power

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

a

b1

,ab

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 ab

b b

0 .

2 2

2

5 5

3 3

Slide 4.1-17

m m

m

a a

b b

.

Simplify.

Solution:

3

30x

x

51

3

3

3

3

x

5

5

1

3

3

27

x

1

243

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

Slide 4.1-18

Using Power Rule (c) CLASSROOM EXAMPLE 6

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

Rules of Exponents

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

Slide 4.1-19

Objective 6

Use combinations of rules.

Slide 4.1-20

4

212

5x

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:

235

3

k

3 42 23xy x y

Slide 4.1-21

Using Combinations of RulesCLASSROOM EXAMPLE 7

Objective 7

Use the rules for exponents in a geometry application.

Slide 4.1-22

Write an expression that represents the area of the figure. Assume x>0.

A LW

2 44 8A x x2 44 8A x

Solution:

632A x 632A x

Slide 4.1-23

Using Area FormulasCLASSROOM EXAMPLE 8