Example 4suggests the following generalizations.
SECTION 5.1 Integer Exponents and Scientific Notation 267
NOW TRY
EXERCISE 5
Apply the quotient rule, if
possible, and write each result
with only positive exponents.
(a) (b)
(c)
m^4
n^3, nZ 0
45
4 -^2
t^8
t^2, tZ 0
Special Rules for Negative ExponentsIf and then and
an
bm
bm
an
.
1
an
aZ 0 bZ 0, an
OBJECTIVE 3 Use the quotient rule for exponents.We simplify a quotient,
such as in much the same way as a product. (In all quotients of this type, assume
that the denominator is not 0.) Consider this example.
Notice that. In the same way, we simplify.
Here 3 - 8 =- 5 .These examples suggest the quotient rule for exponents.
a^3
a^8
=
a#a#a
a#a#a#a#a#a#a#a=
1
a^5
=a-^5
a^38 - 3 = (^5) a 8
a^8
a^3
=
a#a#a#a#a#a#a#a
a#a#a= a#a#a#a#a=a^5
a^8a^3 ,
Quotient Rule for ExponentsIf ais any nonzero real number and mand nare integers, then
That is, when dividing powers of like bases, keep the same base and subtract the
exponent of the denominator from the exponent of the numerator.
am
an
amn.
Using the Quotient Rule for ExponentsApply the quotient rule for exponents, if possible, and write each result with only
positive exponents.
Numerator exponent
Denominator exponent(a) (b)
Subtraction symbol(c) (d)
27
2 -^3
kZ 0 = 27 -^1 -^32 = 27 +^3 = 210
k^7
k^12
= k^7 -^12 =k-^5 =
1
k^5
,
pZ 0
p^6
p^2
=p^6 -^2 =p^4 ,
37
32
= 37 -^2 = 35
EXAMPLE 5
Use parentheses
to avoid errors.(e) (f )
(g) (h) bZ 0
a^3
b^4
zZ 0 ,
z-^5
z-^8
=z-^5 -^1 -^82 =z^3 ,
6
6 -^1
=
61
6 -^1
= 61 -^1 -^12 = 62
8 -^2
85
= 8 -^2 -^5 = 8 -^7 =
1
87
Be careful
with signs.This expression
cannot be simplified
further.The quotient rule does not apply
because the bases are different.
NOW TRYNOW TRY ANSWERS
- (a) (b)
(c)The quotient rule does not
apply.
t^647