268 CHAPTER 5 Exponents, Polynomials, and Polynomial Functions
OBJECTIVE 4 Use the power rules for exponents.We can simplify as
follows.
Notice that. This example suggests the first power rule for exponents.
The other two power rules can be demonstrated with similar examples.
4 # 2 = 8
13422 = 34 # 34 = 34 +^4 = 38
13422
CAUTION Be careful when working with quotients that involve negative expo-
nents in the denominator. Write the numerator exponent, then a subtraction symbol,
and then the denominator exponent. Use parentheses.
NOW TRY
EXERCISE 6
Simplify, using the power
rules.
(a)
(b) a
3 x^2
y^3b3
, yZ 01 - 2 m^324Power Rules for ExponentsIf aand bare real numbers and mand nare integers, then
(a) (b) and (c)
That is,
(a)To raise a power to a power, multiply exponents.
(b) To raise a product to a power, raise each factor to that power.
(c) To raise a quotient to a power, raise the numerator and the denominator to
that power.
a 1 b 02.
a
b
b
mam
bm
1 am 2 namn, 1 ab 2 mambm,
Using the Power Rules for ExponentsSimplify, using the power rules.
(a)
= p^24
= p^8
# 3
1 p^823
EXAMPLE 6
(c)
=
16
81
=
24
34
a
2
3
b
4(b)
= 81 y^4
= 34 y^4
13 y 24
(d)
Multiply exponents.= 36 p^14 Square 6.
= 62 p^14
= 62 p^7 # 2
16 p^722
Power
rule (a) Power
rule (c)Power
rule (b)(e)
= z Z 0 Simplify.
- 8 m^15
z^3
,
=
1 - 223 m^5
3z^3
=
1 - 2 m^52 3
z^3
a
- 2 m^5
z
b
3NOW TRYThe reciprocal of is Also, and are reciprocals.
an#a-n= an#
1
an
= 1
a^1 n= Aa^1 B an a-n
nan.
NOW TRY ANSWERS
- (a) (b)
27 x^6
y^9
16 m^12
Power
rule (b)Power
rule (b)Power
rule (c)