OBJECTIVE 1 Use 0 as an exponent.The definitions of 0 and negative expo-
nents must satisfy the rules for exponents from Section 5.1.For example, if then
and
so that the product rule is satisfied. Check that the power rules are also valid for a 0
exponent. Thus, we define a 0 exponent as follows.
60 # 62 = 1 # 62 = 62 60 # 62 = 60 +^2 = 62 ,
60 =1,
304 CHAPTER 5 Exponents and Polynomials
Zero ExponentFor any nonzero real number a,
Example: 170 = 1
a^0 1.
NOW TRY
EXERCISE 1
Evaluate.
(a)
(b)
(c)
(d) 140 - 120
1 - 12 x 20 1 xZ 02- 120
60
CAUTION Look again at Examples 1(b) and 1(c).In , the base is ,
and since any nonzero base raised to the 0 exponent is 1, In ,
which can be written - 16020 , the base is 60, so - 600 = -1.
1 - 6020 = 1. - 600
1 - 6020 - 60
Using Zero ExponentsEvaluate.
(a) (b)
(c) (d)
(e) (f )
(g) 80 + 110 = 1 + 1 = 2 (h) - 80 - 110 =- 1 - 1 =- 2 NOW TRY
6 y^0 = 6112 = 6 1 yZ 02 16 y 20 = 1 1 yZ 02
- 600 = - 112 = - 1 y^0 = 1 1 yZ 02
600 = 1 1 - 6020 = 1
EXAMPLE 1
OBJECTIVE 2 Use negative numbers as exponents. From the lists at the
beginning of this section, since and we can deduce that should
equal Is the product rule valid in such cases? For example,
The expression behaves as if it were the reciprocal of since their product is 1.
The reciprocal of 62 is also 612 ,leading us to define 6 -^2 as 612.
6 -^262 ,
6 -^2 # 62 = 6 -^2 +^2 = 60 =1.
12 n^.
2 -^3 = 2 - n
12 8 ,
- (^2) =^1
4
NOW TRY ANSWERS
- (a) 1 (b) - 1 (c) 1 (d) 0
Negative ExponentsFor any nonzero real number aand any integer n,
Example: 3 -^2 =
1
32
an
1
an
.
By definition, and are reciprocals, since
Because the definition of can also be written
For example, and a
1
3
b
- 2
6 -^3 = a = 32.
1
6
b
3a-n=
1
an
=
1 n
an
= a
1
a
b
n.
1 n=1, a-n
an#a-n= an#
1
an
= 1.
a-n an
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