Transcript Use the product rule for exponents.

Chapter 5
Section 1
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5.1
1
2
3
4
5
6
7
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  ab   ab .
 
Use combinations of rules.
Use the rules for exponents in a geometric
application.
m
m
m
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Objective 1
Use exponents.
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Slide 5.1 - 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.
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Slide 5.1 - 4
EXAMPLE 1
Using Exponents
Write 2 · 2 · 2 in exponential form and evaluate.
Solution:
222  2  8
3
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Slide 5.1 - 5
EXAMPLE 2
Evaluating Exponential
Expressions
Evaluate. Name the base and the exponent.
Solution:
2
 64
6
1  2  2  2  2  2  2
Base:
 2 
6
 64
2
Exponent:
6
2  2  2  2  2  2
Base
2
Exponent
6
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.
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Slide 5.1 - 6
Objective 2
Use the product rule for
exponents.
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Slide 5.1 - 7
Use the product rule for exponents.
By the definition of exponents,
24  23   2  2  2  2 2  2  2
 2222222
 27
Generalizing from this example
24  23  243  27
suggests the product rule for exponents.
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
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.
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Slide 5.1 - 8
EXAMPLE 3
Using the Product Rule
Use the product rule for exponents to find each product
if possible. Solution:
 7   7 
5
a)
3
  7 
53
  7 
8
58
5
8

4
p
3
p


4

3
p
   
b) 
 12 p13
c)
d)
e)
f)
m  m4
2 5 6
z z z
 m1 4
 z 2  5 6
42  35
64  62
 3888
The product rule does not apply.
 1332
The product rule does not apply.
 m5
 z13
Be sure you understand the difference between adding and multiplying
exponential expressions. For example,
8x3  5x3  8  5 x3  13x3 ,
8x3  5x3   8  5 x33  40 x6 .
but
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Slide 5.1 - 9
Objective 3
Use the rule
m
n
(a )
=
mn
a .
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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.
3 2
8  83 83  833  86
    
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.)
2 4
Example: 3  324  38
 
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Slide 5.1 - 11
EXAMPLE 4
Using Power Rule (a)
Simplify.
Solution:
6

2 5
z 
4 5
 625  610
z
45
z
20
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.
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Slide 5.1 - 12
Objective 4
Use the rule
m
(ab)
=
m
m
a b .
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Slide 5.1 - 13
Use the rule (ab)m = ambm.
We can rewrite the expression (4x)3 as follows.
 4x
3
  4 x  4 x  4 x 
  4  4  4 x  x  x 
 43  x3
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:
2 p
5
 25 p 5
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Slide 5.1 - 14
EXAMPLE 5
Simplify.
Using Power Rule (b)
Solution:
 3a b

 3m
2 3
 3 a
2 4 5
5

2 5
b
 1 3  m

3

4 5

2 3
 243a10b20
 27m6
Use power rule (b) only if there is one term inside parentheses.
Power rule (b) does not apply to a sum. For example,
 4x
2
 42 x 2,
but
 4  x   42  x 2 .
2
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Slide 5.1 - 15
Objective 5
m
Use the rule
a
a
   m.
b
b
m
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Slide 5.1 - 16
m
Use the rule
a
a
   m.
b
b
m
Since the quotient a can be written as a 1 , we use this fact
b
b
and power rule (b) to get power rule (c) for exponents.
m
a
am

For any positive integer m,    m  b  0  .
b
b
(Raise a quotient to a power by raising both numerator and
denominator to the power.)
2
Example:
2
5 5
   2
3 3
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Slide 5.1 - 17
EXAMPLE 6
Simplify.
3
Solution:
3
3
   x  0  x3
 x
1
 
3
5
Using Power Rule (c)
3
27
 3
x
5
1

243
1
 5
3
In general, 1n = 1, for any integer n.
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Slide 5.1 - 18
Rules of Exponents
The rules for exponents discussed in this section are
summarized in the box.
These rules are basic to the study of algebra and should be
memorized.
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Slide 5.1 - 19
Objective 6
Use combinations of rules.
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Slide 5.1 - 20
EXAMPLE 7
Use Combinations of Rules
Simplify
Solution:
2 2
4
4
2
x
2
1
1
   2x   4 
5
5
1

 5k 


 3 
3
2
5 k
2


4 x2

625

3 2
25k 6

9
32
 3xy   x y    1  3  x   y    x   y 
  1 27   x  y  x  y 
2 3
2
4
3
3
3
3
2 3
6
2 4
8
4
4
 27x y
11 10
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Slide 5.1 - 21
Objective 7
Use the rules for exponents in a
geometric application.
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Slide 5.1 - 22
EXAMPLE 8
Using Area Formulas
Find an expression that represents the area of the
figure.
Solution:
A  LW
A  4x
2
8 x 
4
A  48 x
A  32  x
A  32 x
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2 4
6
6
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