Negative Exponents

Chapter
7
Rational Functions
7.1
Negative Exponents
1. Recall what it means to raise a number to a power of −1.
−1
Thus:
=
a −1
b
b
a
An exponent of −1 has nothing to do with
the sign of the answer. Invert a positive
number, you get a positive result. Invert a
negative number, you get a negative result.
=
Thus, to raise a number to a power of −1, simply invert the number.
−1
1
=7
7
3. Recall what it means to raise a number to a power of −1.
−1
Thus:
=
a −1
b
=
b
a
An exponent of −1 has nothing to do with
the sign of the answer. Invert a positive
number, you get a positive result. Invert a
negative number, you get a negative result.
Thus, to raise a number to a power of −1, simply invert the number.
−1
9
8
=−
−
9
8
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CHAPTER 7. RATIONAL FUNCTIONS
414
5. Recall what it means to raise a number to a power of −1.
−1
Thus:
1
a
An exponent of −1 has nothing to do with
the sign of the answer. Invert a positive
number, you get a positive result. Invert a
negative number, you get a negative result.
a−1 =
=
Thus, to raise a number to a power of −1, simply invert the number.
(18)−1 =
1
18
7. Recall what it means to raise a number to a power of −1.
−1
Thus:
1
a
An exponent of −1 has nothing to do with
the sign of the answer. Invert a positive
number, you get a positive result. Invert a
negative number, you get a negative result.
a−1 =
=
Thus, to raise a number to a power of −1, simply invert the number.
(16)−1 =
1
16
9. In this case, we are multiplying like bases, so we’ll use the following law of
exponents:
am an = am+n
. That is, we’ll repeat the base, then add the exponents.
a−9 · a3 = a−9+3
=a
−6
Repeat base, add exponents.
Simplify.
11. In this case, we are multiplying like bases, so we’ll use the following law of
exponents:
am an = am+n
. That is, we’ll repeat the base, then add the exponents.
b−9 · b8 = b−9+8
=b
−1
Second Edition: 2012-2013
Repeat base, add exponents.
Simplify.
7.1. NEGATIVE EXPONENTS
415
13. In this case, we are multiplying like bases, so we’ll use the following law of
exponents:
am an = am+n
. That is, we’ll repeat the base, then add the exponents.
29 · 2−4 = 29+(−4)
=2
5
Repeat base, add exponents.
Simplify.
15. In this case, we are multiplying like bases, so we’ll use the following law of
exponents:
am an = am+n
. That is, we’ll repeat the base, then add the exponents.
9−6 · 9−5 = 9−6+(−5)
=9
−11
Repeat base, add exponents.
Simplify.
17. In this case, we are dividing like bases, so we’ll use the following law of
exponents:
am
= am−n
an
That is, we’ll repeat the base, then subtract the exponents.
26
= 26−(−8)
2−8
= 26+8
=2
14
Repeat base, subtract exponents.
Add the oopposite.
Simplify.
19. In this case, we are dividing like bases, so we’ll use the following law of
exponents:
am
= am−n
an
That is, we’ll repeat the base, then subtract the exponents.
z −1
= z −1−9
z9
= z −1+(−9)
=z
−10
Repeat base, subtract exponents.
Add the opposite.
Simplify.
Second Edition: 2012-2013
CHAPTER 7. RATIONAL FUNCTIONS
416
21. In this case, we are dividing like bases, so we’ll use the following law of
exponents:
am
= am−n
an
That is, we’ll repeat the base, then subtract the exponents.
w−9
= w−9−7
w7
= w−9+(−7)
=w
−16
Repeat base, subtract exponents.
Add the opposite.
Simplify.
23. In this case, we are dividing like bases, so we’ll use the following law of
exponents:
am
= am−n
an
That is, we’ll repeat the base, then subtract the exponents.
7−3
= 7−3−(−1)
7−1
= 7−3+1
=7
−2
Repeat base, subtract exponents.
Add the oopposite.
Simplify.
25. In this case, we are raising a power to a power, so we’ll use the following
law of exponents:
(am )n = amn
That is, we’ll repeat the base, then multiply the exponents.
−1 4
t
= t−1(4)
=t
−4
Repeat base, multiply exponents.
Simplify.
27. In this case, we are raising a power to a power, so we’ll use the following
law of exponents:
n
(am ) = amn
That is, we’ll repeat the base, then multiply the exponents.
−6 7
6
= 6−6(7)
=6
−42
Second Edition: 2012-2013
Repeat base, multiply exponents.
Simplify.
7.1. NEGATIVE EXPONENTS
417
29. In this case, we are raising a power to a power, so we’ll use the following
law of exponents:
n
(am ) = amn
That is, we’ll repeat the base, then multiply the exponents.
−9 −9
z
= z −9(−9)
Repeat base, multiply exponents.
= z 81
Simplify.
31. In this case, we are raising a power to a power, so we’ll use the following
law of exponents:
(am )n = amn
That is, we’ll repeat the base, then multiply the exponents.
−2 3
3
= 3−2(3)
Repeat base, multiply exponents.
= 3−6
Simplify.
−1
. They are equivalent because the
33. Note that 4−3 is equivalent to 43
laws of exponents instruct us to multiply the exponents when raising a power
−1
to another power. To evaluate 43
, we first cube, then invert the result.
−1
Repeat base and multiply exponents.
4−3 = 43
= 64−1
1
=
64
Cube: 43 = 64.
Invert.
Mental approach. It is much easier to simplify this expression mentally. In
the expression 4−3 , the 3 means cube and the minus sign in front of the 3
means “invert.” To do this problem in our head, start with 4 and cube to get
64, then invert to get 1/64.
−1
. They are equivalent because the
35. Note that 2−4 is equivalent to 24
laws of exponents instruct us to multiply the exponents when raising a power
−1
to another power. To evaluate 24
, we first raise to the fourth power, then
invert the result.
−1
2−4 = 24
Repeat base and multiply exponents.
= 16−1
1
=
16
Raise to the fourth power: 24 = 16.
Invert.
Mental approach. It is much easier to simplify this expression mentally. In
the expression 2−4 , the 4 means raise to the fourth power and the minus sign
in front of the 4 means “invert.” To do this problem in our head, start with 2
and raise to the fourth power to get 16, then invert to get 1/16.
Second Edition: 2012-2013
CHAPTER 7. RATIONAL FUNCTIONS
418
37. Note:
−5
1
2
is equivalent to
−1
5
1
2
They are equivalent because the laws of exponents instruct us to multiply the
exponents when raising a power to another power. To evaluate [(1/2)5 ]−1 , we
first raise to the fifth power, then invert the result.
−5 5 −1
1
1
=
2
2
−1
1
=
32
= 32
Repeat base and multiply exponents.
Raise to the fifth power: (1/2)5 = 1/32.
Invert.
Mental approach. It is much easier to simplify this expression mentally. In
−5
the expression (1/2) , the 5 means raise to the fifth power and the minus sign
in front of the 5 means “invert.” To do this problem in our head, start with
1/2 and raise to the fifth power to get 1/32, then invert to get 32.
39. Note:
−5
1
−
2
is equivalent to
5 −1
1
−
2
They are equivalent because the laws of exponents instruct us to multiply the
exponents when raising a power to another power. To evaluate [(−1/2)5 ]−1 ,
we first raise to the fifth power, then invert the result.
−5 5 −1
1
1
=
−
−
2
2
−1
1
= −
32
= −32
Repeat base and multiply exponents.
Raise to the fifth power: (−1/2)5 = −1/32.
Invert.
Mental approach. It is much easier to simplify this expression mentally. In
−5
the expression (−1/2) , the 5 means raise to the fifth power and the minus
sign in front of the 5 means “invert.” To do this problem in our head, start
with −1/2 and raise to the fifth power to get −1/32, then invert to get −32.
Second Edition: 2012-2013
7.1. NEGATIVE EXPONENTS
419
41. All the operators involved are multiplication, so the commutative and associative properties of multiplication allow us to change the order and grouping.
We’ll show this regrouping here, but this step can be done mentally.
−6 −9 8 −8 4u v
5u v
= [(4)(5)](u−6 u8 )(v −9 v −8 )
Multiply 4 and 5 to get 20, then repeat the bases and add the exponents.
= 20u−6+8 v −9+(−8)
= 20u2 v −17
In the solution above, we’ve probably shown way too much work. It’s far easier
to perform all of these steps mentally, multiplying the 4 and the 5 to get 20,
then repeating bases and adding exponents, as in:
−6 −9 8 −8 5u v
= 20u2 v −17
4u v
43. All the operators involved are multiplication, so the commutative and associative properties of multiplication allow us to change the order and grouping.
We’ll show this regrouping here, but this step can be done mentally.
−6 −5 −4x4 y −2 = [(6)(−4)](x−6 x4 )(y −5 y −2 )
6x y
Multiply 6 and −4 to get −24, then repeat the bases and add the exponents.
= −24x−6+4 y −5+(−2)
= −24x−2 y −7
In the solution above, we’ve probably shown way too much work. It’s far easier
to perform all of these steps mentally, multiplying the 6 and the −4 to get −24,
then repeating bases and adding exponents, as in:
−6 −5 −4x4 y −2 = −24x−2 y −7
6x y
45. The simplest approach is to first write the expression as a product.
−6 x7
z9
−6 x7 z 9
·
=
·
4 x−9 z −2
4 x−9 z −2
Reduce −6/4 to lowest terms. Because we are dividing like bases, we repeat
the bases and subtract the exponents.
3
= − x7−(−9) z 9−(−2)
2
3
= − x7+9 z 9+2
2
3
= − x16 z 11
2
Second Edition: 2012-2013
CHAPTER 7. RATIONAL FUNCTIONS
420
In the solution above, we’ve shown way too much work. It’s far easier to
imagine writing the expression as a product, reducing −6/4, then repeating
bases and subtracting exponents, as in:
−6 x7 z 9
3
= − x16 z 11
4 x−9 z −2
2
47. The simplest approach is to first write the expression as a product.
−6 a9 c6
−6 a9
c6
· −5 · −7
=
−5
−7
−4 a c
−4 a
c
Reduce −6/(−4) to lowest terms. Because we are dividing like bases, we repeat
the bases and subtract the exponents.
3 9−(−5) 6−(−7)
a
c
2
3
= a9+5 c6+7
2
3
= a14 c13
2
=
In the solution above, we’ve shown way too much work. It’s far easier to imagine
writing the expression as a product, reducing −6/(−4), then repeating bases
and subtracting exponents, as in:
−6 a9 c6
3
= a14 c13
−4 a−5 c−7
2
49. The law of exponents (ab)n = an bn says that when you raise a product to
a power, you must raise each factor to that power. So we begin by raising each
factor to the power −5.
−5 4 −5
−2 4 −5
w
= 2−5 v −2
2v w
To raise 2 to the −5, first raise 2 to the fifth power, then invert: 2−5 = 1/32.
Next, raising a power to a power requires that we repeat the base and multiply
exponents.
= 1/32 v −2(−5) w4(−5)
= 32 v 10 w−20
In the solution above , we’ve shown way too much work. It’s far easier to
raise each factor to the fifth power mentally: 2−5 = 1/32, then multiply each
exponent on the remaining factors by 5, as in
−2 4 −5
= 1/32 v 10 w−20
2v w
Second Edition: 2012-2013
7.1. NEGATIVE EXPONENTS
421
51. The law of exponents (ab)n = an bn says that when you raise a product to
a power, you must raise each factor to that power. So we begin by raising each
factor to the fourth power.
4 7 4
−1 7 4
y
= 34 x−1
3x y
Note that 34 = 81. Next, raising a power to a power requires that we repeat
the base and multiply exponents.
= 81 x−1(4) y 7(4)
= 81 x−4 y 28
In the solution above , we’ve shown way too much work. It’s far easier to raise
each factor to the fourth power mentally: 34 = 81, then multiply each exponent
on the remaining factors by 4, as in
−1 7 4
= 81 x−4 y 28
3x y
53. The law of exponents (ab)n = an bn says that when you raise a product to
a power, you must raise each factor to that power. So we begin by raising each
factor to the fifth power.
5 −7 5
6 −7 5
= 25 x6
z
2x z
Note that 25 = 32. Next, raising a power to a power requires that we repeat
the base and multiply exponents.
= 32 x6(5) z −7(5)
= 32 x30 z −35
In the solution above , we’ve shown way too much work. It’s far easier to raise
each factor to the fifth power mentally: 25 = 32, then multiply each exponent
on the remaining factors by 5, as in
6 −7 5
= 32 x30 z −35
2x z
55. The law of exponents (ab)n = an bn says that when you raise a product to
a power, you must raise each factor to that power. So we begin by raising each
factor to the power −4.
−4 8 −4
−4 8 −4
= 2−4 a−4
c
2a c
Second Edition: 2012-2013
CHAPTER 7. RATIONAL FUNCTIONS
422
To raise 2 to the −4, first raise 2 to the fourth power, then invert: 2−4 = 1/16.
Next, raising a power to a power requires that we repeat the base and multiply
exponents.
= 1/16 a−4(−4) c8(−4)
= 16 a16 c−32
In the solution above , we’ve shown way too much work. It’s far easier to
raise each factor to the fourth power mentally: 2−4 = 1/16, then multiply each
exponent on the remaining factors by 4, as in
−4 8 −4
= 1/16 a16 c−32
2a c
57. Multiply numerator and denominator by y 2 .
x5 y −2
x5 y −2 y 2
=
· 2
3
z
z3
y
x5 y 0
y2z 3
x5
= 2 3
y z
=
Multiply numerator and
denominator by y 2
Simplify: y −2 y 2 = y 0
Simplify: y 0 = 1
59. Multiply numerator and denominator by s2 .
r9 s−2
r9 s−2 s2
=
· 2
3
t
t3
s
r 9 s0
s 2 t3
r9
= 2 3
s t
=
Multiply numerator and
denominator by s2
Simplify: s−2 s2 = s0
Simplify: s0 = 1
61. Multiply numerator and denominator by y 8 .
x3
x3
y8
=
·
y −8 z 5
y −8 z 5 y 8
x3 y 8
y0z 5
x3 y 8
= 5
z
=
Second Edition: 2012-2013
Multiply numerator and
denominator by y 8
Simplify: y −8 y 8 = y 0
Simplify: y 0 = 1
7.1. NEGATIVE EXPONENTS
423
63. Multiply numerator and denominator by v 4 .
u9
u9
v4
=
·
v −4 w7
v −4 w7 v 4
Multiply numerator and
denominator by v 4
u9 v 4
v 0 w7
u9 v 4
=
w7
Simplify: v −4 v 4 = v 0
=
Simplify: v 0 = 1
65. Multiply 7 and −7 to get −49, then repeat the base and add the exponents.
(7x−1 )(−7x−1 ) = −49x−2
The negative exponent means invert, so we can replace x−2 with 1/x2 , then
multiply numerators and denominators.
−49 1
· 2
1
x
−49
= 2
x
=
67. Multiply 8 and 7 to get 56, then repeat the base and add the exponents.
(8a−8 )(7a−7 ) = 56a−15
The negative exponent means invert, so we can replace a−15 with 1/a15 , then
multiply numerators and denominators.
56 1
·
1 a15
56
= 15
a
=
69. Write the expression as a product.
4 x−9
4x−9
= · 3
3
8x
8 x
Reduce 4/8 to lowest terms, then repeat the base and subtract the exponents.
=
1 −12
·x
2
The negative exponent means invert, so we can replace x−12 with 1/x12 , then
multiply numerators and denominators.
1 1
·
2 x12
1
= 12
2x
=
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CHAPTER 7. RATIONAL FUNCTIONS
424
71. Write the expression as a product.
6c2
6 c2
·
=
−4c7
−4 c7
Reduce 6/(−4) to lowest terms, then repeat the base and subtract the exponents.
3
= − · c−5
2
The negative exponent means invert, so we can replace c−5 with 1/c5 , then
multiply numerators and denominators.
3 1
=− · 5
2 c
3
=− 5
2c
73. First raise each factor to the −4 power.
(−3s9 )−4 = (−3)−4 (s9 )−4
Now, raise −3 to the fourth power, then invert to get 1/81. Next, because we
are raising a power to a power, repeat the base and multiply the exponents.
=
1 −36
s
81
The negative exponent means invert, so we can replace s−36 with 1/s36 , then
multiply numerators and denominators.
1
1
·
81 s36
1
=
81s36
=
75. First raise each factor to the −5 power.
(2y 4 )−5 = 2−5 (y 4 )−5
Now, raise 2 to the fifth power, then invert to get 1/32. Next, because we are
raising a power to a power, repeat the base and multiply the exponents.
=
Second Edition: 2012-2013
1 −20
y
32