Thus, since both and are reciprocals of , the following is true.
Some examples of this result are
and a
1
3
b
- 2
6 -^3 = a = 32.
1
6
b
3a-n= a
1
a
b
nA a-n an
1aB
nSECTION 5.1 Integer Exponents and Scientific Notation 269
NOW TRY
EXERCISE 7
Write with only positive
exponents, and then evaluate.
a5
3
b- 3
Special Rules for Negative Exponents, ContinuedIf and and nis an integer, then
and
That is, any nonzero number raised to the negative nth power is equal to the
reciprocal of that number raised to the nth power.
a
a
b
b
na
b
a
b
nana.
1
a
b
naZ 0 bZ 0
Using Negative Exponents with FractionsWrite with only positive exponents and then evaluate.
(a)
=
49
9
= a
7
3
b
2a
3
7
b
- 2
EXAMPLE 7
NOW TRYChange the fraction to
its reciprocal and change
the sign of the exponent.The definitions and rules of this section are summarized here.
Definitions and Rules for ExponentsFor all integers mand nand all real numbers aand b, the following rules apply.
Product Rule
Quotient Rule
Zero Exponent
Negative Exponent
Power Rules (a) (b)
(c)
Special Rules
a 1 a, b 02
a
b
b
na
b
a
b
nana 1 a 02
1
a
b
n1 a, b 02
an
bm
bm
an
1 a 02
1
an
an
a 1 b 02
a
b
b
mam
bm
1 am 2 namn 1 ab 2 mambm
an 1 a 02
1
an
a^0 1 1 a 02
1 a 02
am
an
amn
am#anamn
NOW TRY ANSWER
- 12527
(b)
or
125
64 x^3
=
53
43 x^3
,
= a
5
4 x
b
3a xZ 0
4 x
5
b
- 3