(d)
Notice in part (c) that we first evaluate the exponential and then find its negative. In
part (d), the -sign is part of the base,
(e) which is not a real number, since
2 - 100 ,is not a real number. NOW TRY
1 - 1002 3/2= 31 - 100)1/2 43 , 1 - 1002 1/2, or
- 27.
1 - 272 2/3= 31 - 272 1/3 42 = 1 - 322 = 9
SECTION 8.2 Rational Exponents 437
NOW TRY
EXERCISE 2
Evaluate each exponential.
(a) (b)
(c) (d)
(e) 1 - 1252 4/3
- 100 3/2 1 - 1212 3/2
32 2/5 8 5/3
When a rational exponent is negative, the earlier interpretation of negative expo-
nents is applied.
If is a real number, then
am/n 1 a 02.
1
am/n
am/n
am/nEvaluating Exponentials with Negative Rational ExponentsEvaluate each exponential.
(a) 16 - 3/4 =
1
16 3/4
=
1
116 1/4 23
=
1
A 2416 B
3 =
1
23
=
1
8
EXAMPLE 3
(b)
(c) =
9
4
1
4
9
=
1
a2
3
b= 2
1
¢
B
3
8
27
≤
= 2
1
a8
27
ba 2/3
8
27
b
- 2/3
=
25 - 3/2=
1
25 3/2
=
1
125 1/2 23
=
1
A 225 B
3 =
1
53
=
1
125
The denominator of 3/4 is
the index and the numerator
is the exponent.Take the reciprocal only of
the base, notthe exponent.1
4
9= 1 ,^49 = 1 #^94
We can also use the rule here, as follows.
a
8
27
b
- 2/3
= a
27
8
b
2/3= a
B
3
27
8
b
2= a
3
2
b
2=
9
4
A
baB
- m
= A
abB
mCAUTION Be careful to distinguish between exponential expressions like the
following.
, which equals which equals and which equals
A negative exponent does not necessarily lead to a negative result. Negative expo-
nents lead to reciprocals, which may be positive.
-
1
2
- 16 1/4, -2, - 16 - 1/4,
1
2
16 - 1/4 ,
NOW TRYNOW TRY
EXERCISE 3
Evaluate each exponential.
(a) (b)
(c) a
216
125
b- 2/3
243 - 3/5 4 - 5/2
NOW TRY ANSWERS
- (a) 4 (b) 32 (c)
(d)It is not a real number.
(e) 625 - (a) 271 (b) 321 (c)^2536
- 1000