OBJECTIVE 2 Define 0 and negative exponents.Consider the following,
where the product rule is applied to an exponent that is not a natural number.
For the product rule to hold, must equal 1, so we define this way for any nonzero
real number a.
40 a^0
42 # 40 = 42 +^0 = 42
SECTION 5.1 Integer Exponents and Scientific Notation 265
NOW TRY
EXERCISE 2
Evaluate.
(a) (b)
(c) - 50 (d) 100 - 90
50 1 - 5 x 20 , xZ 0
CAUTION Be careful not to multiply the bases. In Example 1(a),
not 911 .Keep the same base and add the exponents.
34 # 37 = 311 ,
Zero ExponentIf ais any nonzero real number, then
a^0 1.
The expression 00 is undefined.*
*In advanced treatments, 00 is called an indeterminate form.Using 0 as an ExponentEvaluate.
(a) (b)
(c) (d)
(e)
= 2
= 1 + 1
50 + 120
- 60 =- 1602 =- 112 =- 1 - 1 - 620 =- 1
60 = 1 1 - 620 = 1
EXAMPLE 2
(f ) 18 k 20 =1, kZ 0
NOW TRYThe base is 6,
not-6.Here the
base is-6.To define a negative exponent, we extend the product rule, as follows.
Here, is the reciprocal of. But is the reciprocal of and a number can have
only one reciprocal. Therefore, 8 -^2 = 812 .We can generalize this result.
8 -^28281282 ,
82 # 8 -^2 = 82 +^1 -^22 = 80 = 1
Negative ExponentFor any natural number nand any nonzero real number a,
an
1
an
.
With this definition, the expression is meaningful for any integer exponent n
and any nonzero real number a.
an
NOW TRY ANSWERS
- (a) 1 (b) 1 (c) - 1 (d) 0