(d)
Power rule (b)= - 1 # 518 Power rule (a)
= 1 - 123 # 15623
= 1 - 1 # 5623 - a=- 1 #a
1 - 5623
SECTION 5.1 The Product Rule and Power Rules for Exponents 299
= - 518 NOW TRY
CAUTION Power rule ( b) does not apply to a sum.For example,
but
OBJECTIVE 5 Use the rule .Since the quotient can be written as
aA^1 bB,we use this fact and power rule (b) to get power rule (c) for exponents.
a
A b
a
bBma
m
bm14 x 22 = 42 x^2 , 14 +x 22 Z 42 + x^2.
Power Rule (c) for ExponentsFor any positive integer m,
(Raise a quotient to a power by raising both numerator and denominator to the
power.)
Example: a
5
3
b
2=
52
32
a 1 b 02.
a
b
b
mam
bm
NOW TRY
EXERCISE 6
Simplify.
(a) (b)
1 qZ 02a1
4
b3
ap
qb5NOW TRY ANSWERS
- (a) (b)
- (a) (b) 641
p^5
q^5- 125 a^3 b^348 t^6 p^10
Using Power Rule (c)Use power rule (c) for exponents to simplify.
(a) (b)
(c) a 14 = 1 # 1 # 1 # 1 = 1 NOW TRY
1
5
b
4=
14
54
=
1
54
=
1
625
a 1 nZ 02
m
n
b
3=
m^3
n^3
a
2
3
b
5=
25
35
=
32
243
EXAMPLE 6
NOTE In Example 6(c),we used the fact that.
In general, 1 n1, for any integer n.
14 = 1
Rules for ExponentsFor positive integers mand n, the following are true.
Examples
Product rule
Power rules (a)
(b)
(c) a
5
3
b
2=
52
32
a 1 b 02
a
b
b
mam
bm
1 ab 2 mambm 1 2 p 25 = 25 p^5
13224 = 32
# 4
1 am 2 namn = 38
am#anamn 62 # 65 = 62 +^5 = 67
Raise
to the
designated
power.- 1
NOW TRY
EXERCISE 5
Simplify.
(a) 1 - 5 ab 23 (b) 314 t^3 p^522