Changing from Negative to Positive ExponentsSimplify by writing with positive exponents. Assume that all variables represent
nonzero real numbers.
(a) (b)
(c)
a-^2 b
3 d-^3
=
bd^3
3 a^2
m-^5
p-^1
=
p^1
m^5
=
p
m^5
4 -^2
5 -^3
=
53
42
=
125
16
EXAMPLE 3
306 CHAPTER 5 Exponents and Polynomials
NOW TRY
EXERCISE 3
Simplify by writing with
positive exponents. Assume
that all variables represent
nonzero real numbers.
(a) (b)
(c)
x^2 y-^3
5 z-^4m^2 n-^45 -^3
6 -^2
(d) a NOW TRY
x
2 y
b
- 4
= a
2 y
x
b
4=
24 y^4
x^4
=
16 y^4
x^4
OBJECTIVE 3 Use the quotient rule for exponents.Consider the following.
The difference between the exponents, is the exponent in the quotient.
Also,
Here, 2 - 4 =- 2 .These examples suggest the quotient rule for exponents.
62
64
=
6 # 6
6 # 6 # 6 # 6=
1
62
= 6 -^2.
5 - 3 = 2 ,
65
63
=
6 # 6 # 6 # 6 # 6
6 # 6 # 6= 62
Notice that bin the numerator and 3 in
the denominator are not affected.NOW TRY ANSWERS
- (a) (b)
(c)
x^2 z^4
5 y^3m^2
n^462
53
, or
36
125CAUTION Be careful. We cannot use this rule to change negative exponents to
positive exponents if the exponents occur in a sum or differenceof terms. For example,
would be written with positive exponents as.
1
52
+
1
3
7 -
1
23
5 -^2 + 3 -^1
7 - 2 -^3
Quotient Rule for ExponentsFor any nonzero real number aand any integers mand n,
(Keep the same base and subtract the exponents.)
Example:
58
54
= 58 -^4 = 54
am
an
amn.
CAUTION A common erroris to write By the quotient rule,
the quotient must have the same base,5, just as in the product rule.
If you are not sure, use the definition of an exponent to write out the factors.
58
54
=
5 # 5 # 5 # 5 # 5 # 5 # 5 # 5
5 # 5 # 5 # 5= 54
58
54
= 58 -^4 = 54
5854 =^1
8 - (^4) = 14.
http://www.ebook777.com
http://www.ebook777.com