Download 4.6 Positive Integral Exponents

4.6
4.6
In this
section
Positive Integral Exponents
237
POSITIVE INTEGRAL EXPONENTS
The product rule for positive integral exponents was presented in Section 4.2, and
the quotient rule was presented in Section 4.5. In this section we review those rules
and then further investigate the properties of exponents.
The Product and Quotient
Rules
The Product and Quotient Rules
●
Raising an Exponential
Expression to a Power
The rules that we have already discussed are summarized below.
●
Power of a Product
●
Power of a Quotient
●
Summary of Rules
●
(4-31)
The following rules hold for nonnegative integers m and n and a 0.
a m a n a mn
am
n a mn if m n
a
am
1
if n m
n nm
a
a
a0 1
Product rule
Quotient rule
Zero exponent
CAUTION
The product and quotient rules apply only if the bases of the
expressions are identical. For example, 32 34 36, but the product rule cannot be
applied to 52 34. Note also that the bases are not multiplied: 32 34 96.
Note that in the quotient rule the exponents are always subtracted, as in
x7
x4
x3
and
y5
1
8 3.
y
y
If the larger exponent is in the denominator, then the result is placed in the
denominator.
E X A M P L E
1
Using the product and quotient rules
Use the rules of exponents to simplify each expression. Assume that all variables
represent nonzero real numbers.
b) (3x)0(5x2)(4x)
a) 23 22
2
8x
(3a2b)b9
c) 5
d) 2x
(6a5)a3b2
Solution
a) Because the bases are both 2, we can use the product rule:
Product rule
23 22 25
32
Simplify.
0
2
2
b) (3x) (5x )(4x) 1 5x 4x Definition of zero exponent
20x 3
Product rule
2
8x
4
c) 5 3
Quotient rule
2x
x
238
(4-32)
Chapter 4
Polynomials and Exponents
d) First use the product rule to simplify the numerator and denominator:
study
tip
Keep track of your time for
one entire week. Account for
how you spend every half
hour. Add up your totals for
sleep, study, work, and recreation. You should be sleeping
50–60 hours per week and
studying 1–2 hours for every
hour you spend in the classroom.
(3a2b)b9
3a2b10
5 3 2 (6a )a b
6a8b2
b8
6
2a
Product rule
Quotient rule
■
Raising an Exponential Expression to a Power
When we raise an exponential expression to a power, we can use the product rule to
find the result, as shown in the following example:
(w4)3 w4 w4 w4
w12
Three factors of w4 because of the exponent 3
Product rule
By the product rule we add the three 4’s to get 12, but 12 is also the product of 4 and 3.
This example illustrates the power rule for exponents.
Power Rule
If m and n are nonnegative integers and a 0, then
(a m)n a mn.
In the next example we use the new rule along with the other rules.
E X A M P L E
2
Using the power rule
Use the rules of exponents to simplify each expression. Assume that all variables
represent nonzero real numbers.
(23)4 27
3(x 5)4
b) c) a) 3x 2(x 3)5
5
9
2 2
15x 22
Solution
a) 3x 2(x 3)5 3x 2x15
3x17
3 4
7
212 27
(2 ) 2
b) 214
25 2 9
219
214
25
32
5 4
3(x )
3x 20
1
c) 22 15x
15x 22 5x 2
Power rule
Product rule
Power rule and product rule
Product rule
Quotient rule
Evaluate 25.
■
Power of a Product
Consider an example of raising a monomial to a power. We will use known rules to
rewrite the expression.
(2x)3 2x 2x 2x
222xxx
23x3
Definition of exponent 3
Commutative and associative properties
Definition of exponents
4.6
(4-33)
Positive Integral Exponents
239
Note that the power 3 is applied to each factor of the product. This example illustrates the power of a product rule.
Power of a Product Rule
If a and b are real numbers and n is a positive integer, then
(ab)n a nb n.
E X A M P L E
3
Using the power of a product rule
Simplify. Assume that the variables are nonzero.
b) (3m)3
a) (xy3)5
c) (2x 3y 2z7)3
Solution
a) (xy 3)5 x 5(y 3)5 Power of a product rule
x 5y15
Power rule
3
3 3
b) (3m) (3) m Power of a product rule
27m3 (3)(3)(3) 27
c) (2x 3y 2z7)3 23(x 3)3(y 2)3(z7)3 8x9y6z 21
■
Power of a Quotient
helpful
hint
Note that these rules of exponents are not absolutely
necessary. We could simplify
every expression here by
using only the definition of
exponent. However, these
rules make it a lot simpler.
Raising a quotient to a power is similar to raising a product to a power:
x 3 x x x
Definition of exponent 3
5
5 5 5
xxx
Definition of multiplication of fractions
555
x3
3
Definition of exponents
5
The power is applied to both the numerator and denominator. This example illustrates the power of a quotient rule.
Power of a Quotient Rule
If a and b are real numbers, b 0, and n is a positive integer, then
a
b
E X A M P L E
4
n
an
n.
b
Using the power of a quotient rule
Simplify. Assume that the variables are nonzero.
2 2
3x4 3
a) 3
b) 3
5x
2y
Solution
2 2
22
a) 3 5x
(5x 3)2
4
6
25x
Power of a quotient rule
(5x 3)2 52(x 3)2 25x 6
12a5b
c) 4a2b7
3
240
(4-34)
Chapter 4
Polynomials and Exponents
b)
3x4
3
2y
33x12
23y9
27x12
8y 9
3
Power of a quotient and power of a product rules
Simplify.
c) Use the quotient rule to simplify the expression inside the parentheses before
using the power of a quotient rule.
12a5b
4a2b7
3a3
b6
3
27a9
b18
3
Use the quotient rule first.
Power of a quotient rule
■
Summary of Rules
helpful
hint
Note that the rules of exponents show how exponents
behave with respect to multiplication and division only. We
studied the more complicated
problem of using exponents
with addition and subtraction
in Section 4.4 when we
learned rules for (a b)2 and
(a b)2.
The rules for exponents are summarized in the following box.
Rules for Nonnegative Integral Exponents
The following rules hold for nonzero real numbers a and b and nonnegative
integers m and n.
1. a0 1 Definition of zero exponent
2. am an amn Product rule
am
3. n amn for m n,
a
am
1
for n m
n a
anm
Quotient rule
4. (am)n amn Power rule
5. (ab)n an bn Power of a product rule
a
6. b
WARM-UPS
n
an
n
b
Power of a quotient rule
True or false? Assume that all variables represent nonzero real
numbers. A statement involving variables is to be marked true only if it
is an identity. Explain your answer.
1. 30 1 False
4. (2x)4 2x4 False
7. (ab3)4 a4b12
2y3
10. 9
2
4y6
81
2. 25 28 413 False
5. (q3)5 q8 False
a12
True 8. 4 a3 False
a
True
3. 23 33 65 False
6. (3x 2)3 27x6 False
6w4
9. 9 2w5 False
3w
4.6
4. 6
(4-35)
Positive Integral Exponents
241
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
3. Why must the bases be the same in these rules?
These rules do not make sense without identical bases.
4. What is the power rule for exponents?
The power rule for exponents says that (am)n amn.
5. What is the power of a product rule?
The power of a product rule says that (ab)n anbn.
6. What is the power of a quotient rule?
b.
n
n
a
n
For all exercises in this section, assume that the variables represent nonzero real numbers.
Simplify the exponential expressions. See Example 1.
8. x6 x7 x13
9. (3u8)(2u2)
6u10
10. (3r4)(6r 2) 18r 6
11. a3b4 ab6(ab)0 a4b10
12. x2y x3y6(x y)0
x5y7
3t9 1
14. 6t18 2t9
3xy 5xy
16. 20x3y14
8
9
18. 2 10
2
3
3
3y
4x
2a3 1
13. 4a7 2a4
2a 5b 3a7b3
15. 15a6b8
2a6
4
5b
4000
19. (x )
x
6
20. (y )
(t 2)5
23. 34
(t )
2x12
3x(x5)2
25. 6x3(x 2)4
y
8
22. (y2)6 3y5
(r4)2
24. 53
(r )
1
2
t
3y17
1
7
r
Simplify. See Example 3.
27. (xy2)3 x3y6
29. (2t5)3 8t15
31. (2x2y5)3
8x6y15
(a4b2c5)3 9 2 14
33. abc
a3b4c
28. (wy2)6 w6y12
30. (3r3)3 27r9
32. (3y2z3)3 27y6z9
(2ab2c3)5 2b6c11
34. 34 (2a bc)
a7
4
24
2
4
9 4 3
12
6 4 3
6
10
8
3
2 4 9 2
y6
8
5
6
2
8
21
2 7
9
Simplify each expression. Your answer should be an integer or a
fraction. Do not use a calculator.
43. 32 62 45
44. (5 3)2 4
45. (3 6)2
46. 52 32 16
81
47. 23 33 19
49. (2 3)
51.
5
2
3
48. 33 43 91
1
3
50. (3 4)3
8
125
52.
4
3
3
343
27
64
53. 52 23 200
54. 103 33 27,000
55. 23 24 128
56. 102 104 1,000,000
3 2
2
5
2
1
16
58.
Simplify each expression.
59. 3x4 5x7 15x11
3
3
3
2
1
81
60. 2y3(3y)
125x12
61. (5x4)3
62. (4z3)3
6y4
64z9
63. 3y z 9yz 27y z
64. 2a b 2a9b2 4a13b7
9u4v9
65. 3u5v8
20a5b13
66. 4
5a b13
7
6 19
3v
u
8a b
71. 4c
8x y
73. 16x y
2 3
5y3(y5)2
1
26. 5
4
2 6
10y (y ) 2y
1
2
3
2
3
y2
2
3 2
8
67. (xt2)(2x2t)4 16x9t 6
2 4
21. 2x 2 (x2)5
36.
12
5 12
17. 23 52 200
9r
81r
38. t
t
81y
3y
40. 16z
2zy
10rs t
125s
42. 2rs t
t
x12
64
3
57.
Simplify. See Example 2.
2 3
3
x4
4
2 4
2. What is the quotient rule for exponents?
The quotient rule says that aman amn if m n and
1
aman nm
if n m.
a
7. 22 25 128
2a
16a
37. b
b
2x y
x
39. 4y
8y
6x y z
4z
41. 3x y z
x
35.
1. What is the product rule for exponents?
The product rule says that a man a mn.
a
The power of a quotient rule says that b
Simplify. See Example 4.
69.
2x
x4
8
6
x
3 4 5
5
4 7
5 6
5
4 5
68. (ab)3(3ba2)4 81a11b7
9y
10a c
72. 5a b
5x yz
74. 5x yz
70.
32a15b20
25
c
y5
5
32x
4a
3y8
y5
2
6
5
5
5 4
2
2
32c5
2
b0
3 5
z10
Solve each problem.
75. Long-term investing. Sheila invested P dollars at annual
rate r for 10 years. At the end of 10 years her investment
was worth P(1 r)10 dollars. She then reinvested this
money for another 5 years at annual rate r. At the end
of the second time period her investment was worth
242
(4-36)
Chapter 4
Polynomials and Exponents
P(1 r)10 (1 r)5 dollars. Which law of exponents can be
used to simplify the last expression? Simplify it.
P(1 r)15
76. CD rollover. Ronnie invested P dollars in a 2-year CD
with an annual rate of return of r. After the CD rolled over
two times, its value was P((1 r)2)3. Which law of exponents can be used to simplify the expression? Simplify it.
P(1 r)6
4.7
In this
section
GET TING MORE INVOLVED
77. Writing. When we square a product, we square each factor
in the product. For example, (3b)2 9b2. Explain why we
cannot square a sum by simply squaring each term of the
sum.
78. Writing. Explain why we define 20 to be 1. Explain why
20 1.
NEGATIVE EXPONENTS AND
SCIENTIFIC NOTATION
We defined exponential expressions with positive integral exponents in Chapter 1
and learned the rules for positive integral exponents in Section 4.6. In this section
you will first study negative exponents and then see how positive and negative integral exponents are used in scientific notation.
●
Negative Integral Exponents
●
Rules for Integral Exponents
Negative Integral Exponents
●
Converting from Scientific
Notation
●
Converting to Scientific
Notation
If x is nonzero, the reciprocal of x is written as 1. For example, the reciprocal of 23 is
x
written as 13. To write the reciprocal of an exponential expression in a simpler way,
●
Computations with
Scientific Notation
2
1
we use a negative exponent. So 23 23. In general we have the following definition.
Negative Integral Exponents
If a is a nonzero real number and n is a positive integer, then
1
an .
(If n is positive, n is negative.)
an
E X A M P L E
1
Simplifying expressions with negative exponents
Simplify.
a) 25
calculator
close-up
You can evaluate expressions
with negative exponents on a
calculator as shown here.
b) (2)5
2 3
c) 2
3
Solution
1
1
a) 25 5 2
32
1
b) (2)5 5 Definition of negative exponent
(2)
1
1
32
32
23
c) 2 23 32
3
1
1
3 2
2
3
1 1 1 9 9
8 9 8 1 8
■