Using Negative ExponentsSimplify by writing with positive exponents. Assume that all variables represent
nonzero real numbers.
(a) (b)
(c) and 2 are reciprocals.
Notice that we can change the base to its reciprocal if we also change the sign of
the exponent.
(d) (e)
(f ) 4 -^1 - 2 -^1 =
1
4
-
1
2
=
1
4
-
2
4
=-
1
4
a
4
3
b
- 5
= a
3
4
b
5=
243
1024
a
2
5
b
- 4
= a
5
2
b
4=
625
16
1a 2
1
2
b
- 3
= 23 = 8
5 -^3 =
1
53
=
1
125
3 -^2 =
1
32
=
1
9
EXAMPLE 2
SECTION 5.2 Integer Exponents and the Quotient Rule 305
NOW TRY
EXERCISE 2
Simplify.
(a) (b)
(c) (d)
(e)p-^41 pZ 02
a 3 -^2 + 4 -^23
2
b- 4
a1
7
b- 2
2 -^3
Changing from Negative to Positive ExponentsFor any nonzero numbers aand band any integers mand n, the following are true.
and
Examples: and a
4
5
b
- 3
= a
5
4
b
3 -^53
2 -^4
=
24
35
a
a
b
b
ma
b
a
b
am m
bn
bn
am
(^25) and are reciprocals. (^52)
Apply the exponents first,
and then subtract.
(g)
(h)
Power rule (c)and xare reciprocals.(i) NOW TRY
Consider the following.
Therefore,
2 -^3
3 -^4
=
34
23
.
1
23
#^3
41
=
34
23
=
1
23
,
1
34
=
1
23
1
34
2 -^3
3 -^4
=
x^3 y-^4 =
x^3
y^4
1=x x
4= a
1
x
b
- 4
1
x-^4
=
1 -^4
x-^4
p-^2 =
1
p^2
Remember to find a
common denominator.It is convenient to write 1 as
here, because is the
exponent in the denominator.1 -^4 - 4To divide by a fraction,
multiply by its reciprocal.NOW TRY ANSWERS
- (a) (b) 49 (c)
(d) (e)
1
p^425
14416
811
8