chapter 5 section 1. objectives 1 copyright © 2012, 2008, 2004 pearson education, inc. the product...
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Chapter 5 Section 1
Objectives
1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 rule
Use combinations of rules.
Use the rules for exponents in a geometric application.
.m m
m
a a
b b
5.1
2
3
4
5
6
7
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Use exponents.
Slide 5.1-3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Write 2 · 2 · 2 in exponential form and evaluate.
Solution:
2 2 2 832
Slide 5.1-5
EXAMPLE 1 Using Exponents
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 indicate that the exponent applies only to the base 2, not −2.
Slide 5.1-6
EXAMPLE 2 Evaluating Exponential Expressions
62
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 2
Use the product rule for exponents.
Slide 5.1-7
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 5.1-8
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 5.1-9
EXAMPLE 3 Using the Product Rule
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 3
Use the rule (am)n = amn.
Slide 5.1-10
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 5.1-11
Use the rule (am)n = amn.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 5.1-12
EXAMPLE 4 Using Power Rule (a)
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Objective 4
Use the rule (ab)m = am bm.
Slide 5.1-13
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 5.1-14
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 5.1-15
EXAMPLE 5 Using Power Rule (b)
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Objective 5 m m
m
a ab b
Use the rule .
Slide 5.1-16
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 5.1-17
m m
m
a a
b b
.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 5.1-18
EXAMPLE 6 Using Power Rule (c)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 5.1-19
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 6
Use combinations of rules.
Slide 5.1-20
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 5.1-21
EXAMPLE 7 Using Combinations of Rules
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 7
Use the rules for exponents in a geometry application.
Slide 5.1-22
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 5.1-23
EXAMPLE 8 Using Area Formulas
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Homework
• Pg 301-302 # 16-90 Even
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