basic

1. Chapter 5Chapter 5
Section 1Section 1
1

2. 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
= am
bn
.
Use the rule
Use combinations of rules.
Use the rules for exponents in a geometric
application.
1
4
3
2
6
5
5.15.1
7
 
= ÷
 
.
m m
m
a a
b b

3. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 11
Slide 5.1 - 3
Use exponents.

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

5. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1
Write 2 · 2 · 2 in exponential form and evaluate.
Solution:
Using Exponents
Slide 5.1 - 5
2 2 2× × 8=3
2=

6. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Evaluate. Name the base and the exponent.
EXAMPLE 2
Evaluating Exponential
Expressions
Slide 5.1 - 6
6
2−
Solution:
64= −
( )
6
2− 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.

7. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 22
Slide 5.1 - 7
Use the product rule for
exponents.

8. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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, am
· an
= am + n
.
(Keep the same base; add the exponents.)
Example: 62
· 65
= 67
( ) ( )4 3
2 2 2 2 2 2 2 2 2× = × × × × ×
4 3 4 3 7
2 2 2 2+
× = =
2 2 2 2 2 2 2= × × × × × ×
7
2=
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
.

9. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3
Solution:
Using the Product Rule
Slide 5.1 - 9
( ) ( )
5 3
7 7− −
( ) 5 8
4 3 p +
= − ×
( )
5 3
7
+
= − ( )
8
7= −
13
12p= −
Use the product rule for exponents to find each product
if possible.
a)
b)
c)
d)
e)
f)
( )( )5 8
4 3p p−
4
m m×
2 5 6
z z z
2 5
4 3×
4 2
6 6+
2 5 6
z + +
=
1 4
m +
= 5
m=
13
z=
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,
but
( )3 3 3 3
8 5 8 5 3 ,1x x x x++ = =
( )( ) ( )3 3 3 3 6
8 5 8 5 4 .0x x x x+
=×=

10. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 33
Slide 5.1 - 10
Use the rule (am
)n
= amn
.

11. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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 6
8 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 8
3 3 3×
= =

12. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4
Solution:
10
6=2 5
6 ×
=
20
z=
( )
52
6
4 5
z ×
=
Using Power Rule (a)
Slide 5.1 - 12
Simplify.
( )
54
z
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.

13. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 44
Use the rule (ab)m
= am
bm
.
Slide 5.1 - 13

14. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use the rule (ab)m
= am
bm
.
We can rewrite the expression (4x)3
as follows.
Slide 5.1 - 14
( ) ( ) ( ) ( )
3
4 4 4 4x x x x=
( ) ( )4 4 4 x x x= × × × ×
3 3
4 x= ×
This example suggests power rule (b) for exponents.
For any positive integer m, (ab)m
= am
bm
.
(Raise a product to a power by raising each factor to the power.)
Example: ( )
5 5 5
2 2p p=

15. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5
Simplify.
Solution:
( )
52 4
3a b
Using Power Rule (b)
Slide 5.1 - 15
( )
32
3m−
( ) ( )
5 55 2 4
3 a b=
( )
33 2
1 3 m= − ×
10 20
243a b=
6
27m= −
Power rule (b) does not apply to a sum. For example,
, but( )
2 2 2
4 4x x= ( )
2 2 2
4 4 .x x+ ≠ +
Use power rule (b) only if there is one term inside parentheses.

16. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 55
Use the rule
Slide 5.1 - 16
m m
m
a a
b b
 
= ÷
 
.

17. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
b
1
,a
b
( )
m m
m
a a
b
b b
 
= ≠ ÷
 
0 .
2 2
2
5 5
3 3
 
= ÷
 

18. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 6
Simplify.
Solution:
( )
3
3
0x
x
 
≠ ÷
 
Using Power Rule (c)
Slide 5.1 - 18
5
1
3
 
 ÷
 
3
3
3
x
=
5
5
1
3
=
3
27
x
=
1
243
=
In general, 1n
= 1, for any integer n.

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

20. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 66
Use combinations of rules.
Slide 5.1 - 20

21. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 7
( )
4
21
2
5
x
 
 ÷
 
Slide 5.1 - 21
Simplify
( )
22 3
2
5
3
k×
=
6
25
9
k
=
( ) ( ) ( ) ( ) ( ) ( )
3 43 3 3 42 2
1 3 x y x y= − ×
( )2 24
4
21
5 1
x
= ×
2
4
625
x
=
( ) ( ) ( )( )( )( )3 6 8 4
1 27 x y x y= −
11 10
27x y= −
Solution:
Use Combinations of Rules
23
5
3
k 
 ÷
 
( ) ( )
3 42 2
3xy x y−

22. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 77
Use the rules for exponents in a
geometric application.
Slide 5.1 - 22

23. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 8Using Area Formulas
Slide 5.1 - 23
Find an expression that represents the area of the
figure.
A LW=
( )( )2 4
4 8A x x=
2 4
4 8A x +
= × ×
Solution:
6
32A x= ×
6
32A x=

24. Chapter 5Chapter 5
Section 1Section 1
24
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Do you find these slides were useful?
If Yes ,Join Dreams School “Campaign
for Female Education”
Help us in bringing a change in a girl life,
because “When someone takes away your
pens you realize quite how important
education is”.
. Just Click on any advertisement on the
page, your one click can make her smile
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