Print this page Chapter 6 Rules for Exponents and the Reasons for Them 6.1 INTEGER POWERS AND THE EXPONENT RULES Repeated addition can be expressed as a product. For example, Similarly, repeated multiplication can be expressed as a power. For example, Here, 2 is called the base and 5 is called the exponent. Notice that 2 5 is not the same as 5 2 , because 2 5 = 2 × 2 × 2 × 2 × 2 = 32 but 5 2 = 5 × 5 = 25. In general, if a is any number and n is a positive integer, then we define Notice that a 1 = a, because here we have only 1 factor of a. For example, 5 1 = 5. We call a 2 the square of a and a 3 the cube of a. Multiplying and Dividing Powers with the Same Base When we multiply powers with the same base, we can add the exponents to get a more compact form. For example, 5 2 · 5 3 = (5 · 5) · (5 · 5 · 5) = 5 2 + 3 = 5 5 . In general, Thus, Example 1 Write with a single exponent: (a) q 5 · q 7 (b) 6 2 · 6 3 (c) 2 n · 2 m (d) 3 n · 3 4 1 of 22
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Chapter Rules for Exponents and the Reasons for Them...6 Rules for Exponents and the Reasons for Them 6.1 INTEGER POWERS AND THE EXPONENT RULES Repeated addition can be expressed as
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Chapter
6 Rules for Exponents and theReasons for Them
6.1 INTEGER POWERS AND THE EXPONENT RULESRepeated addition can be expressed as a product. For example,
Similarly, repeated multiplication can be expressed as a power. For example,
Here, 2 is called the base and 5 is called the exponent. Notice that 25 is not the same as 52, because25 = 2 × 2 × 2 × 2 × 2 = 32 but 52 = 5 × 5 = 25.
In general, if a is any number and n is a positive integer, then we define
Notice that a1 = a, because here we have only 1 factor of a. For example, 51 = 5. We call a2 the square of a and a3 thecube of a.
Multiplying and Dividing Powers with the Same Base
When we multiply powers with the same base, we can add the exponents to get a more compact form. For example,52 · 53 = (5 · 5) · (5 · 5 · 5) = 52 + 3 = 55. In general,
Thus,
Example 1 Write with a single exponent:
(a) q5 · q7
(b) 62 · 63
(c) 2n · 2m
(d) 3n · 34
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(e) (x + y)2(x + y)3.
SolutionUsing the rule an · am = an + m we have
(a) q5 · q7 = q5 + 7 = q12
(b) 62 · 63 = 62 + 3 = 65
(c) 2n · 2m = 2n + m
(d) 3n · 34 = 3n + 4
(e) (x + y)2(x + y)3 = (x + y)2 + 3 = (x + y)5.
Just as we applied the distributive law from left to right as well as from right to left, we can use the rulean · am = an + m written from right to left as an + m = an · am.
Example 2 Write as a product:
(a) 52 + a
(b) xr + 4
(c) yt + c
(d) (z + 2)z + 2.
SolutionUsing the rule an + m = an · am we have
(a) 52 + a = 52 · 5a = 25 · 5a
(b) xr + 4 = xr · x4
(c) yt + c = yt · yc
(d) (z + 2)z + 2 = (z + 2)z · (z + 2)2.
When we divide powers with a common base, we subtract the exponents. For example, when we divide 56 by 52, weget
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More generally, if n > m,
Thus,
Example 3 Write with a single exponent:
(a)
(b)
(c) , where n > 4
(d)
(e) .
SolutionSince we have
(a)
(b)
(c)
(d)
(e) .
Just as with the products, we can write in reverse as .
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Example 4 Write as a quotient:
(a) 102 - k
(b) eb - 4
(c) zw - s
(d) (p + q)a - b.
SolutionSince we have
(a)
(b)
(c)
(d)
Raising a Power to a Power
When we take a number written in exponential form and raise it to a power, we multiply the exponents. For example,
More generally,
Thus,
Example 5 Write with a single exponent:
(a) (q7)5
(b) (7p)3
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(c) (ya)b
(d) (2x)x
(e)
(f) .
SolutionUsing the rule (am)n = am · n we have
(a) (q7)5 = q7 · 5 = q35
(b) (7p)3 = 73p
(c) (ya)b = yab
(d)
(e)
(f) .
Example 6 Write as a power raised to a power:
(a) 23 · 2
(b) 43x
(c) e4t
(d) .
SolutionUsing the rule am · n = (am)n we have
(a) 23 · 2 = (23)2. This could also have been written as (22)3.(b) 43x = (43)x, which simplifies to 64x. This could also have been written as (4x)3.(c) e4t = (e4)t. This could also have been written as (et)4.(d)
Products and Quotients Raised to the Same Exponent
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When we multiply 52 · 42 we can change the order of the factors and rewrite it as52 · 42 = (5 · 5) · (4 · 4) = 5 · 5 · 4 · 4 = (5 · 4) · (5 · 4) = (5 · 4)2 = 202. Sometimes, we want to use this process inreverse: 102 = (2 · 5)2 = 22 · 52.
In general,
Thus,
Example 7 Write without parentheses:
(a) (qp)7
(b) (3x)n
(c) (4ab2)3
(d) (2x2n)3n.
SolutionUsing the rule (ab)n = anbn we have
(a) (qp)7 = q7p7
(b) (3x)n = 3nxn
(c) (4ab2)3 = 43a3(b2)3 = 64a3b6
(d) .
Example 8 Write with a single exponent:
(a) c4d4
(b) 2n · 3n.(c) 4x2
(d) a4(b + c)4
(e) (x2 + y2)5(c - d)5.
SolutionUsing the rule anbn = (ab)n we have
(a) c4d4 = (cd)4
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(b) 2n · 3n = (2 · 3)n = 6n
(c) 4x2 = 22x2 = (2x)2
(d) a4(b + c)4 = (a(b + c))4
(e) .
Division of two powers with the same exponent works the same way as multiplication. For example,
Or, reversing the process,
More generally,
Thus,
Example 9 Write without parentheses:
(a)
(b)
(c)
(d) .
SolutionUsing the rule we have
(a)
(b)
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(c)
(d) .
Example 10 Write with a single exponent:
(a)
(b)
(c)
(d)
(e)
SolutionUsing the rule we have
(a)
(b)
(c)
(d)
(e) .
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Zero and Negative Integer Exponents
We have seen that 45 means 4 multiplied by itself 5 times, but what is meant by 40, 4-1 or 4-2? We choose definitionsfor exponents like 0, -1, -2 that are consistent with the exponent rules.
If a ≠ 0, the exponent rule for division says
But , so we define a0 = 1 if a ≠ 0. The same idea tells us how to define negative powers. If a ≠ 0, the exponent
rule for division says
But , so we define a-1 = 1/a. In general, we define
Note that a negative exponent tells us to take the reciprocal of the base and change the sign of the exponent, not tomake the number negative.
Example 11 Evaluate:
(a) 50
(b) 3-2
(c) 2-1
(d) (-2)-3
(e)
Solution (a) Any nonzero number to the zero power is one, so 50 = 1.(b) We have
(c) We have
(d) We have
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(e) We have
With these definitions, we have the exponent rule for division, where n and m are integers.
Example 12 Rewrite with only positive exponents. Assume all variables are positive.
(a)
(b)
(c)
(d)
Solution (a) We have
(b) We have
(c) We have
(d) We have
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In part (a) of Example 12, we saw that the x-2 in the denominator ended up as x2 in the numerator. In general:
Example 13 Write each of the following expressions with only positive exponents. Assume all variablesare positive.
(a)
(b)
(c)
(d)
(e)
Solution (a) .
(b) .
(c) .
(d) .
(e) .
Summary of Exponent Rules
We summarize the results of this section as follows.
general
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Expressions with a Common Base
If m and n are integers,
1. an · am = an + m
2.
3. (am)n = am · n
Expressions with a Common Exponent
If n is an integer,
1. (ab)n = anbn
2.
Zero and Negative Exponents
If a is any nonzero number and n is an integer, then:
• a0 = 1•
Common Mistakes
Be aware of the following notations that are sometimes confused:
For example, -24 = -(24) = -16, but (-2)4 = (-2)(-2)(-2)(-2) = 16.
Example 14 Evaluate the following expressions for x = -2 and y = 3:
Evaluate the expressions in Exercises 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 without using a calculator.
1. 3 · 23
Answer:
24
2. -32
3. (-2)3
Answer:
-8
4. 51 · 14 · 32
5. 52 · 22
Answer:
100
6.
7. (-5)3 · (-2)2
Answer:
13 of 22
-500
8. -53 · -22
9. -14 · (-3)2(-23)
Answer:
72
10.
11. 30
Answer:
1
12. 03
In Exercises 13, 14, 15, 16, 17, 18, 19, 20, 21 and 22, evaluate the following expressions for x = 2, y = -3, andz = -5.
13. -xyz
Answer:
-30
14. yx
15. -yx
Answer:
-9
16.
17.
Answer:
-8/125
18. x-z
19. -x-z
Answer:
-32
20.
14 of 22
21.
Answer:
729/1000
22.
In Exercises 23, 24, 25, 26, 27, 28, 29, 30 and 31 , write the expression in the form xn, assuming x ≠ 0.
23. x3 · x5
Answer:
x8
24.
25. (x4 · x)2
Answer:
x10
26.
27.
Answer:
x2
28.
29. (x3)5
Answer:
x15
30.
31.
Answer:
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x5
In Exercises 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44 and 45, write with a single exponent.
32. 42 · 4n
33. 2n22
Answer:
2n + 2
34. a5b5
35.
Answer:
(a/b)x
36.
37.
Answer:
22n - m
38. An + 3BnB3
39. BaBa + 1
Answer:
B2a + 1
40. (x2 + y)3(x + y2)3
41.
Answer:
(x + y)20
42. 162y8
43.
Answer:
(g + h)3
44.
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45.
Answer:
(a + b)3
Without a calculator, decide whether the quantities in Exercises 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58and 59 are positive or negative
46. (-4)3
47. -43
Answer:
Negative
48. (-3)4
49. -34
Answer:
Negative
50. (-23)42
51. -3166
Answer:
Negative
52. 17-1
53. (-5)-2
Answer:
Positive
54. -5-2
55. (-4)-3
Answer:
Negative
56. (-73)0
57. -480
Answer:
Negative
58. (-47)-15
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59. (-61)-42
Answer:
Positive
In Exercises 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71 and 72, write each expression without parentheses.Assume all variables are positive.
60.
61.
Answer:
c12/d4
62.
63.
Answer:
64r6/125s12
64.
65.
Answer:
36g10/49h14
66. (cf)9
67. (2p)5
Answer:
32p5
68.
69.
Answer:
18 of 22
16tb4t
70.
71.
Answer:
3 · 16x e4x
72.
PROBLEMS
In Problems 73, 74, 75, 76 and 77, decide which expressions are equivalent. Assume all variables are positive.
73. (a) 3-2
(b)
(c)
(d)
(e)
Answer:
(a), (c), (d) equivalent; (b), (e) equivalent
74. (a)
(b)
(c)
(d)
(e)
75. (a)
(b)
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(c)
(d) x-r
(e)
Answer:
(a), (c), (e) equivalent; (b), (d) equivalent
76. (a)
(b)
(c)
(d)
(e)
77. (a)
(b)
(c)
(d)
(e)
Answer:
(a), (b) equivalent; (c), (d), (e) equivalent
In Problems 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90 and 91 , write each expression as a product or aquotient. Assume all variables are positive.
78. 32 + 3
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79. a4 + 1
Answer:
a4 + 1 = a4 · a
80. e2 + r
81. 104 - z
Answer:
104/10z
82. ka - b
83. 4p + 3
Answer:
4p · 43
84. 6a - 1
85. (-n)a + b
Answer:
(-n)a(-n)b
86. xa + b + 1
87. p1 - (a + b)
Answer:
p/(papb)
88. (r - s)t + z
89. (p + q)a - b
Answer:
(p + q)a/(p + q)b
90. et - 1(t + 1)
91. (x + 1)ab + c
Answer:
(x + 1)ab(x + 1)c
In Problems 92, 93, 94, 95, 95, 96, 97 and 98, write each expression as a power raised to a power. There may bemore than one correct answer.