7.7
Negative Integer Exponents
7.7
OBJECTIVES
1. Define the zero exponent
2. Use the definition of a negative exponent to
simplify an expression
3. Use the properties of exponents to simplify
expressions that contain negative exponents
In Section 1.4, all the exponents we looked at were positive integers. In this section, we
look at the meaning of zero and negative integer exponents. First, let’s look at an application of the quotient rule that will yield a zero exponent.
Recall that, in the quotient rule, to divide expressions with the same base, keep the base
and subtract the exponents.
am
amn
an
Now, suppose that we allow m to equal n. We then have
am
amm a0
am
(1)
But we know that it is also true that
am
1
am
(2)
Comparing equations (1) and (2), we see that the following definition is reasonable.
Rules and Properties:
NOTE We must have a 0. The
form 00 is called indeterminate
and is considered in later
mathematics classes.
The Zero Exponent
For any real number a when a 0,
a0 1
Example 1
The Zero Exponent
Use the above definition to simplify each expression.
NOTE Notice that in 6x0 the
(a) 170 1
(b) (a3b2)0 1
(c) 6x0 6 1 6
(d) 3y0 3
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exponent 0 applies only to x.
CHECK YOURSELF 1
Simplify each expression.
(a) 250
(b) (m4n2)0
(c) 8s0
(d) 7t0
Recall that, in the product rule, to multiply expressions with the same base, keep the
base and add the exponents.
am an am+n
555
556
CHAPTER 7
RATIONAL EXPRESSIONS AND FUNCTIONS
Now, what if we allow one of the exponents to be negative and apply the product rule?
Suppose, for instance, that m 3 and n 3. Then
am an a3 a3 a3(3)
a0 1
so
a3 a3 1
Dividing both sides by a3, we get
a3 1
a3
Rules and Properties:
Negative Integer Exponents
NOTE John Wallis (1616–1702),
For any nonzero real number a and whole number n,
an English mathematician, was
the first to fully discuss the
meaning of 0, negative, and
rational exponents.
an 1
an
and an is the multiplicative inverse of an.
Example 2 illustrates this definition.
Example 2
Using Properties of Exponents
NOTE From this point on, to
Simplify the following expressions.
simplify will mean to write the
expression with positive
exponents only.
(a) y5 1
y5
(b) 42 1
1
42
16
variables so that they represent
nonzero real numbers.
(c) (3)3 (d)
2
3
3
1
1
1
3 (3)
27
27
1
2
3
3
1
27
8
8
27
CHECK YOURSELF 2
Simplify each of the following expressions.
(a) a10
(b) 24
(c) (4)2
(d)
5
2
2
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NOTE Also, we will restrict all
NEGATIVE INTEGER EXPONENTS
SECTION 7.7
557
Example 3 illustrates the case in which coefficients are involved in an expression with
negative exponents. As will be clear, some caution must be used.
Example 3
Using Properties of Exponents
Simplify each of the following expressions.
CAUTION
The expressions
4w2
and
(4w)2
are not the same. Do you see
why?
(a) 2x3 2 1
2
3 3
x
x
The exponent 3 applies only
to the variable x, and not to the
coefficient 2.
(b) 4w2 4 (c) (4w)2 1
4
2 w
w2
1
1
(4w)2
16w2
CHECK YOURSELF 3
Simplify each of the following expressions.
(a) 3w4
(b) 10x5
(c) (2y)4
(d) 5t2
Suppose that a variable with a negative exponent appears in the denominator of an expression. Our previous definition can be used to write a complex fraction that can then be
simplified. For instance,
1
1
a2
a2
1
a2
1
1
a2
Negative exponent
in denominator.
Positive exponent in numerator.
To divide, we
invert and multiply.
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To avoid the intermediate steps, we can write that, in general,
Rules and Properties:
Negative Exponents in a Denominator
For any nonzero real number a and integer n,
1
an
an
558
CHAPTER 7
RATIONAL EXPRESSIONS AND FUNCTIONS
Example 4
Using Properties of Exponents
Simplify each of the following expressions.
(a)
1
y3
y3
(b)
1
25 32
25
(c)
3
3x2
4x2
4
(d)
b4
a3
b4
a3
The exponent 2 applies only to x, not to 4.
CHECK YOURSELF 4
Simplify each of the following expressions.
(a)
NOTE To review these
properties, return to Section 1.4.
1
x4
(b)
1
33
(c)
2
3a2
(d)
c5
d 7
The product and quotient rules for exponents apply to expressions that involve any integer exponent—positive, negative, or 0. Example 5 illustrates this concept.
Example 5
Using Properties of Exponents
Simplify each of the following expressions, and write the result, using positive exponents
only.
x4 (b)
m5
m5(3) m53
m3
m2 (c)
NOTE Notice that m5 in the
numerator becomes m5 in the
denominator, and m3 in the
denominator becomes m3 in the
numerator. We then simplify as
before.
1
x4
Add the exponents by the
product rule.
Subtract the exponents by the
quotient rule.
1
m2
x5 (3)
x2
x5x3
x2 (7) x9
x7
x7
x7
We apply first the product rule
and then the quotient rule.
In simplifying expressions involving negative exponents, there are often alternate
approaches. For instance, in Example 5(b), we could have made use of our earlier work to
write
m3
1
m5
m35 m2 2
3 m
m5
m
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(a) x3 x7 x3(7)
NEGATIVE INTEGER EXPONENTS
SECTION 7.7
559
CHECK YOURSELF 5
Simplify each of the following expressions.
(a) x9 x5
(b)
y7
y3
(c)
a3a2
a5
The properties of exponents can be extended to include negative exponents. One of
these properties, the quotient-power rule, is particularly useful when rational expressions
are raised to a negative power. Let’s look at the rule and apply it to negative exponents.
Rules and Properties:
b
a
n
a
, b0
bn
Rules and Properties:
b
a
Quotient-Power Rule
n
n
n
n
a
b
b
n
bn
a
a
Raising Quotients to a Negative Power
n
a 0, b 0
Example 6
Extending the Properties of Exponents
Simplify each expression.
s3
t2
2
(a)
(b)
m2
n2
3
t2
s3
2
n2
m2
t4
s6
3
n6
1
6 6 6
m
nm
CHECK YOURSELF 6
Simplify each expression.
(a)
s3
3t 2
3
(b)
x5
y2
3
As you might expect, more complicated expressions require the use of more than one of
the properties for simplification. Example 7 illustrates such cases.
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Example 7
Using Properties of Exponents
Simplify each of the following expressions.
(a)
(a2)3(a3)4
a6 a12
(a3)3
a9
a612
a6
a9
a9
a6(9) a69 a15
Apply the power rule to each factor.
Apply the product rule.
Apply the quotient rule.
CHAPTER 7
RATIONAL EXPRESSIONS AND FUNCTIONS
NOTE It may help to separate
(b)
the problem into three
fractions, one for the
coefficients and one for each of
the variables.
(c)
8x2y5
8 x2 y5
12x4y3
12 x4 y3
2 2(4) 53
x
y
3
2 2 8
2x2
x y 8
3
3y
pr3s5
p3r3s2
2
p3r3s2
pr3s5
p6r6s4
p2r6s10
2
p4r12s6 CAUTION
p4s6
r12
Be Careful! Another possible first step (and generally an efficient one) is to rewrite an
expression by using our earlier definitions.
an 1
an
and
1
an
an
For instance, in Example 8(b), we would correctly write
8x2y5
8x4
12x4y3
12x2y3y5
A common error is to write
12x4
8x2y5
12x4y3
8x2y3y5
This is not correct.
The coefficients should not be moved along with the factors in x. Keep in mind that the
negative exponents apply only to the variables. The coefficients remain where they were in
the original expression when the expression is rewritten by using this approach.
CHECK YOURSELF 7
Simplify each of the following expressions.
(a)
(x5)2(x2)3
(x4)3
(b)
12a3b2
16a2b3
(c)
xy3z5
x4y2z3
CHECK YOURSELF ANSWERS
1
1
1
4
; (c) ; (d)
10 ; (b)
a
16
16
25
2
3
10
1
5
2a
d7
4
;
(b)
27;
(c)
;
(d)
3. (a) 4 ; (b) 5 ; (c)
;
(d)
4.
(a)
x
w
x
16y4
t2
3
c5
6
1
27t
1
3
y3z24
5. (a) x4; (b) 4 ; (c) a4
6. (a) 9 ; (b) 15 6
7. (a) x8; (b)
;
(c)
y
s
x y
4ab5
x15
1. (a) 1; (b) 1; (c) 8; (d) 7
2. (a)
3
© 2001 McGraw-Hill Companies
560
Name
Exercises
7.7
Section
Date
In exercises 1 to 22, simplify each expression.
1. x5
2. 33
ANSWERS
3. 52
4. x8
5. (5)2
6. (3)3
7. (2)3
8. (2)4
9.
10.
3
4
2
11. 3x2
13. 5x4
14. (2x)4
16. 5x2
17.
19.
2
5x3
20.
2
3
3
12. 4x3
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
15. (3x)2
1
x3
18.
3
4x4
21.
1
x5
x3
y4
5
22.
x
y3
In exercises 23 to 32, use the properties of exponents to simplify the expressions.
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23. x5 x3
24. y4 y5
25. a9 a6
26. w5 w3
27. z2 z8
28. b7 b1
29. a5 a5
30. x4 x4
31.
32.
x5
x2
x3
x6
561
ANSWERS
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
In exercises 33 to 58, use the properties of exponents to simplify the following.
33. (x5)3
34. (w4)6
35. (2x3)(x2)4
36. (p4)(3p3)2
37. (3a4)(a3)(a2)
38. (5y2)(2y)(y5)
39. (x4y)(x2)3(y3)0
40. (r4)2(r2s)(s3)2
41. (ab2c)(a4)4(b2)3(c3)4
42. (p2qr2)(p2)(q3)2(r2)0
43. (x5)3
44. (x2)3
45. (b4)2
46. (a0b4)3
47. (x5y3)2
48. (p3q2)2
49. (x4y2)3
50. (3x2y2)3
51. (2x3y0)5
52.
a6
b4
53.
x2
y4
x3
y2
55.
x4
y2
56.
(3x4)2(2x2)
x6
44.
43.
45.
46.
47.
48.
49.
50.
51.
52.
54.
53.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
54.
3
57. (4x2)2(3x4)
67.
58. (5x4)4(2x3)5
562
59. (2x5)4(x3)2
60. (3x2)3(x2)4(x2)
61. (2x3)3(3x3)2
62. (x2y3)4(xy3)0
63. (xy5z)4(xyz2)8(x6yz)5
64. (x2y2z2)0(xy2z)2(x3yz2)
65. (3x2)(5x2)2
66. (2a3)2(a0)5
67. (2w3)4(3w5)2
© 2001 McGraw-Hill Companies
In exercises 59 to 90, simplify each expression.
ANSWERS
3 2
3x6 y5
69.
2y9 x3
4 5
68. (3x ) (2x )
2
5 6 4
71. (7x y)(3x y )
74. (5a2b4)(2a5b0)
77.
80.
15x3y2z4
20x4y3z2
xy3z4
x y z
3 2 2
72.
75.
78.
2
81.
2w5z3
3x3y9
68.
x8 2y9
70. 6 3
y
x
x5y4
w4z0
(x3)(y2)
y3
24x5y3z2
36x2y3z2
x2y2 x4y2
x3y2 x2y2
69.
70.
2
2 3
4 2
73. (2x y )(3x y )
71.
76.
79.
82.
6x3y4
24x2y2
x5y7
x0y4
x3y3
x4y2
3
x2y2
xy4
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
1
82.
n3
83. x2n x3n
84. xn1 x3n
85.
x
xn1
83.
84.
86.
xn4
xn1
87. (y n)3n
88. (xn1)n
85.
86.
89.
x2n xn2
x3n
90.
xn x3n5
x4n
87.
88.
91. Can (a b)1 be written as
not? Explain.
1
1
by using the properties of exponents? If not, why
a
b
92. Write a short description of the difference between (4)3, 43, (4)3, and 43.
89.
90.
91.
Are any of these equal?
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92.
93. If n 0, which of the following expressions are negative?
(n)3, n3, n3, (n)3, (n)3, n3
93.
If n 0, which of these expressions are negative? Explain what effect a negative in
the exponent has on the sign of the result when an exponential expression is
simplified.
563
Answers
1.
15.
1
x5
3.
1
9x2
1
25
17. x3
29. 1
33. x15
31.
y4
x2
65. 75x2
y5
x3
2
87. y3n
1
x3
43.
55.
7. 2x3
5
y2
x4
48
x8
67. 144w2
3xy5
4z6
89. x2
77.
69.
79.
91.
27
8
y4
x3
11.
23. x2
3
x2
13.
25.
1
a3
5
x4
27.
1
z10
39. x10y
45. b8
57.
9.
21.
37. 3a
1
x15
1
8
x10
y6
47.
59. 16x26
3x3
2y4
1
5 3
xy
49. x12y6
61.
72
x3
71. 567x22y25
81.
y8
x7
83. x5n
51.
x15
32
63. x42y33z25
73.
6
x2y5
85. x2
93.
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75.
19.
35. 2x5
41. a17b8c13
53.
1
25
5.
564