Download 5.2 THE POWER RULES

268
(5-12)
Chapter 5
Exponents and Polynomials
5.2
In this
section
●
Raising an Exponential
Expression to a Power
●
Raising a Product to a
Power
●
Raising a Quotient to a
Power
●
Variable Exponents
●
Summary of the Rules
●
Applications
THE POWER RULES
In Section 5.1 you learned some of the basic rules for working with exponents.
All of the rules of exponents are designed to make it easier to work with exponential expressions. In this section we will extend our list of rules to include three
new ones.
Raising an Exponential Expression to a Power
An expression such as (x 3)2 consists of the exponential expression x 3 raised to the
power 2. We can use known rules to simplify this expression.
(x 3)2 x 3 x 3
x6
Exponent 2 indicates two factors of x3.
Product rule: 3 3 6
Note that the exponent 6 is the product of the exponents 2 and 3. This example
illustrates the power of a power rule.
Power of a Power Rule
If m and n are any integers and a 0, then
(am)n amn.
E X A M P L E
1
calculator
close-up
A graphing calculator cannot
prove that the power of a
power rule is correct, but it
can provide numerical support for it.
Using the power of a power rule
Use the rules of exponents to simplify each expression. Write the answer with
positive exponents only. Assume all variables represent nonzero real numbers.
b) (x2)6
a) (23)5
(x 2)1
c) 3(y3)2y5
d) (x3)3
Solution
a) (23)5 215
Power of a power rule
2 6
12
b) (x ) x
Power of a power rule
1
12
Definition of a negative exponent
x
c) 3(y3)2y5 3y6y5 Power of a power rule
3y
Product rule
2 1
2
(x )
x
d) Power of a power rule
(x 3)3 x9
x7
Quotient rule
■
5.2
The Power Rules
(5-13)
269
Raising a Product to a Power
calculator
Consider how we would simplify a product raised to a positive power and a product
raised to a negative power using known rules.
close-up
3 factors of 2x
You can use a graphing calculator to illustrate the power of
a product rule.
(2x) 2x 2x 2x 23 x3 8x3
1
1
1
3 3
(ay)3 3 3 3 a y
(ay)
(ay)(ay)(ay) a y
3
In each of these cases the original exponent is applied to each factor of the product.
These examples illustrate the power of a product rule.
Power of a Product Rule
If a and b are nonzero real numbers and n is any integer, then
(ab)n an bn.
E X A M P L E
2
Using the power of a product rule
Simplify. Assume the variables represent nonzero real numbers. Write the answers
with positive exponents only.
b) (2x2)3
c) (3x2y3)2
a) (3x)4
Solution
a) (3x)4 (3)4x 4 Power of a product rule
81x 4
b) (2x 2)3 (2)3(x 2)3 Power of a product rule
8x 6
Power of a power rule
1
x4
c) (3x2y3)2 (3)2(x2)2( y3)2 x 4y6 6
9
9y
■
Raising a Quotient to a Power
calculator
Now consider an example of applying known rules to a power of a quotient:
close-up
You can use a graphing calculator to illustrate the power of
a quotient rule.
3
x x x
x3
3
5 5 5 5
x
5
We get a similar result with a negative power:
3
x
5
5
x
3
x3
5 5 5 53
3 53
x x x
x
In each of these cases the original exponent applies to both the numerator and
denominator. These examples illustrate the power of a quotient rule.
Power of a Quotient Rule
If a and b are nonzero real numbers and n is any integer, then
a
b
n
an
n.
b
270
(5-14)
Chapter 5
E X A M P L E
3
Exponents and Polynomials
Using the power of a quotient rule
Use the rules of exponents to simplify each expression. Write your answers with
positive exponents only. Assume the variables are nonzero real numbers.
x 3
2x3 3
x2 1
3 2
b) 2
c) d) 3
a) 3
2
3y
2
4x
helpful
hint
The exponent rules in this section apply to expressions that
involve only multiplication
and division. This is not too
surprising since exponents,
multiplication, and division
are closely related. Recall
that
a3 a a a
and
a b a b1.
Solution
x 3 x3
a) 3 Power of a quotient rule
2
2
x3
8
3 3
(2)3x 9
2x
b) 2 3
Because (x 3)3 x9 and (y2)3 y6
3y
3 y6
8x9
8x9
6 6
27y
27y
2 1
2
x
x
3 2 (3)2
42x6
16x6
c) 3 8x 2
d) 3
3
2
6 2 ■
2
2
4x
4 x
(3)
9
A fraction to a negative power can be simplified by using the power of a quotient
rule as in Example 3. Another method is to find the reciprocal of the fraction first, then
use the power of a quotient rule as shown in the next example.
E X A M P L E
4
Negative powers of fractions
Simplify. Assume the variables
with positive exponents only.
3 3
b)
a) 4
are nonzero real numbers and write the answers
x2
5
2
Solution
3 3
4 3
3 4
a) The reciprocal of is .
4 3
4
3
43
3
Power of a quotient rule
3
64
27
2 2
x
5 2
52
25
b) 2 22 4
5
x
(x )
x
c)
2y3
3
2
2y3
c) 3
2
2
3
3
2y
9
6
4y
■
Variable Exponents
So far, we have used the rules of exponents only on expressions with integral
exponents. However, we can use the rules to simplify expressions having variable
exponents that represent integers.
E X A M P L E
5
Expressions with variables as exponents
Simplify. Assume the variables represent integers.
b) (52x )3x
a) 34y 35y
2n
c) m
3
5n
5.2
calculator
Solution
a) 34y 35y 39y
2
b) (52x)3x 56x
(2n)5n
2n 5n
c) m (3m)5n
3
2
25n
35mn
close-up
Did we forget to include the
rule (a b)n an bn? You
can easily check with a calculator that this rule is not correct.
The Power Rules
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271
Product rule: 4y 5y 9y
Power of a power rule: 2x 3x 6x 2
Power of a quotient rule
Power of a power rule
■
Summary of the Rules
The definitions and rules that were introduced in the last two sections are summarized in the following box.
Rules for Integral Exponents
For these rules m and n are integers and a and b are nonzero real numbers.
1
1. an n Definition of negative exponent
a
1
1 n
1
2. an , a1 , and an Negative exponent rules
an
a
a
3. a0 1 Definition of zero exponent
4. a ma n a mn Product rule
am
5. n amn
Quotient rule
a
6. (am)n amn Power of a power rule
7. (ab)n anbn Power of a product rule
a n an
8. n
Power of a quotient rule
b
b
helpful
hint
In this section we use the
amount formula for interest
compounded annually only.
But you probably have money
in a bank where interest is
compounded daily. In this
case r represents the daily rate
(APR365) and n is the number of days that the money is
on deposit.
E X A M P L E
6
Applications
Both positive and negative exponents occur in formulas used in investment situations.
The amount of money invested is the principal, and the value of the principal after a
certain time period is the amount. Interest rates are annual percentage rates.
Amount Formula
The amount A of an investment of P dollars with interest rate r compounded
annually for n years is given by the formula
A P(1 r)n.
Finding the amount
According to Fidelity Investments of Boston, U.S. common stocks have returned
an average of 10% annually since 1926. If your great-grandfather had invested
$100 in the stock market in 1926 and obtained the average increase each year, then
how much would the investment be worth in the year 2006 after 80 years of
growth?
272
(5-16)
Chapter 5
calculator
close-up
With a graphing calculator
you can enter 100(1 0.10)80
almost as it appears in print.
Exponents and Polynomials
Solution
Use n 80, P $100, and r 0.10 in the amount formula:
A P(1 r)n
A 100(1 0.10)80
100(1.1)80
204,840.02
So $100 invested in 1926 would have amounted to $204,840.02 in 2006.
■
When we are interested in the principal that must be invested today to grow to a
certain amount, the principal is called the present value of the investment. We can
find a formula for present value by solving the amount formula for P :
A P(1 r)n
A
P n
(1 r)
P A(1 r)n
Divide each side by (1 r)n.
Definition of a negative exponent
Present Value Formula
The present value P that will amount to A dollars after n years with interest
compounded annually at annual interest rate r is given by
P A(1 r)n.
E X A M P L E
7
Finding the present value
If your great-grandfather wanted you to have $1,000,000 in 2006, then how much
could he have invested in the stock market in 1926 to achieve this goal? Assume he
could get the average annual return of 10% (from Example 6) for 80 years.
Solution
Use r 0.10, n 80, and A 1,000,000 in the present value formula:
P
P
P
P
A(1 r)n
1,000,000(1 0.10)80
1,000,000(1.1)80 Use a calculator with an exponent key.
488.19
A deposit of $488.19 in 1926 would have grown to $1,000,000 in 80 years at a rate
■
of 10% compounded annually.
WARM-UPS
True or false? Explain your answer. Assume all variables represent
nonzero real numbers.
1. (22)3 25 False
4. (23)3 227 False
1
7. 3
2
2
10. x
2
3
2
2
x
True
4 True
2. (23)1 8 True
5. (2x)3 6x 3 False
3. (x3)3 x9 True
6. (3y3)2 9y9 False
8. 3 27
9. 2 8
2
3
8
True
x2
3
x6
True
5.2
5. 2
(5-17)
The Power Rules
273
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is the power of a power rule?
The power of a power rule says that (am)n amn.
2. What is the power of a product rule?
The power of a product rule says that (ab)m ambm.
3. What is the power of a quotient rule?
The power of a quotient rule says that (ab)m ambm.
4. What is principal?
Principal is the amount of money invested initially.
5. What formula is used for computing the amount of an
investment for which interest is compounded annually?
To compute the amount A when interest is compounded annually, use A P(1 i)n, where P is the principal, i is the
annual interest rate, and n is the number of years.
6. What formula is used for computing the present value of an
amount in the future with interest compounded annually?
To compute the present value P for the amount A in n years
at annual interest rate i, use P A(1 i)n.
For all exercises in this section, assume the variables represent
nonzero real numbers and use positive exponents only in your
answers.
Use the rules of exponents to simplify each expression. See
Example 1.
7. (22)3
8. (32)2
9. ( y2)5
64
81
y10
10. (x6)2
x12
13. (m3)6
m18
11. (x2)4
1
8
x
12. (x2)7
1
14
x
14. (a3)3
a9
15. (x2)3(x3)2
1
(a2)3
18. (a 2)4
(x 3)4
16. (m3)1(m2)4 17. 2
(x )5
1
1
5
2
m
x
a2
Simplify. See Example 2.
19. (9y)2 81y2
20. (2a)
21. (5w 3)2 25w6
22.
23. (x 3y2)3
x9
6
y
b2
2
9a
2
2xy
6x3
27. 2
1
(3x y)
y
1
(2ab)2
29. 34
8a b
2ab2
25. (3ab1)2
24.
26.
28.
30.
8a
8
(2w5)3 w15
a4
(a2b3 )2 6
b
x3
(2x1y2)3 6
8y
1
3ab
15a2b
(5ab2)1
1
(3xy)3
81x4y6
3xy3
3
3
Simplify. See Example 3.
w 3 w3
31. 2
8
3a 3
27a3
33. 4
64
2x1 2 x2y2
35. y
4
3 2
3x
y2
37. 6
y
9x
m2
25
2
16
34. 3b
81b
2a b
27
36. 3
8a b
2y
x
38. x
8y
m
32. 5
2
4
4
2
3
6 3
2 3
3
6
Simplify. See Example 4
2 2 25
39. 5
4
2
1
41. 4
2
2x 3
27
43. 3
3
8x
2x2 3 27y3
45. 3y
8x6
2
9
2
9
42. 3 4
ab
c
44. c
ab
ab
46. ab
ab
3
40. 4
16
2
1
3 2
2 8
2
Simplify each expression. Assume that the variables represent
integers. See Example 5.
47. 52t 54t 56t
48. 32n3 342n 3
3w 2w
6w 2
49. (2 )
2
50. 68x (62x)3 62x
2m6
7
43p p
m3
51. m
7
52.
4
7 3
4 4p
53. 82a1 (8a4)3 85a11
54. (543y)3(5y2)2 587y
Use the rules of exponents to simplify each expression. If possible, write down only the answer.
55. 3x4 2x 5
56. (3x4)2
57. (2x 2)3
9
8
6x
9x
8x6
2 1
3x y
21x2
58. 3x 2 2x4
59. 60. 1
z
y 2
x2y2
6
3z
2
2
x
xy
2
1
1
2
1
2x3 2
61. 62. 63. 3
5
3
3
4x6
5
2
9
2y4 3
64. 65. (2x2)1
66. (3x2)3
x
8y12
x2
27
6
3
x
2
x
Use the rules of exponents to simplify each expression.
2x2y 3 y3
2x3y2 1 3y
67. 68. 3
3
2
2
8x
2x
xy
3xy
(5a1b2)3 b14
(2m2n3)4 16m7
69. 70. 1
(5ab2)4 5a7
mn5
n7
(5-18)
Chapter 5
(2x2y)3
71. (2x2y7)
(2xy1)2
x6
8
16y
6a2b3 2
73. (3a1b2)3
2c4
3ac8
Exponents and Polynomials
(3x1y3)2
72. (9x9y5)
(3xy1)3
y2
27x10
7xy1 3
74. (7xz2)4 z
1
7xy3z11
Write each expression as 2 raised to a power. Assume that the
variables represent integers.
75. 32 64
76. 820
77. 81 64
211
260
24
6
6
3n
78. 10 20
79. 4
80. 6n5 35n
26
26n
2n5
n
m
1
32
81. m
82. 16
1283n
2
24m
216n
83.
85.
87.
89.
Use a calculator to evaluate each expression. Round
approximate answers to three decimal places.
1
(2.5)3
84. 6.25
2
25
5
(2.5)5
2 1
86. 21 2
21 22 0.75
3
(0.036)2 (4.29)3
88. 3(4.71)2 5(0.471)3
850.559
18.700
(5.73)1 (4.29)1
90. [5.29 (0.374)1]3
(3.762)1
1.533
505.080
Value of $10,000 investment
(in thousands of dollars)
Solve each problem. See Examples 6 and 7.
91. Deeper in debt. Melissa borrowed $40,000 at 12% compounded annually and made no payments for 3 years. How
much did she owe the bank at the end of the 3 years? (Use
the compound interest formula.) $56,197.12
92. Comparing stocks and bonds. According to Fidelity Investments of Boston, throughout the 1990s annual returns
on common stocks averaged 19%, whereas annual returns
on bonds averaged 9%.
a) If you had invested $10,000 in bonds in 1990 and
achieved the average return, then what would your investment be worth after 10 years in 2000? $23,673.64
b) How much more would your $10,000 investment be
worth in 2000 if you had invested in stocks?
$33,273.20
93. Saving for college. Mr. Watkins wants to have $10,000 in
a savings account when his little Wanda is ready for
college. How much must he deposit today in an account paying 7% compounded annually to have $10,000
in 18 years? $2,958.64
94. Saving for retirement. In the 1990s returns on Treasury
Bills fell to an average of 4.5% per year (Fidelity Investments). Wilma wants to have $2,000,000 when she retires
in 45 years. If she assumes an average annual return of
4.5%, then how much must she invest now in Treasury Bills
to achieve her goal? $275,928.73
95. Life expectancy of white males. Strange as it may seem,
your life expectancy increases as you get older. The function
L 72.2(1.002)a
can be used to model life expectancy L for U.S. white males
with present age a (National Center for Health Statistics,
www.cdc.gov/nchswww).
a) To what age can a 20-year-old white male expect to
live? 75.1 years
b) To what age can a 60-year-old white male expect
to live? (See also Chapter Review Exercises 153 and
154.) 81.4 years
96. Life expectancy of white females. Life expectancy improved more for females than for males during the 1940s
and 1950s due to a dramatic decrease in maternal mortality
rates. The function
L 78.5(1.001)a
can be used to model life expectancy L for U.S. white
females with present age a.
a) To what age can a 20-year-old white female expect to
live? 80.1 years
b) Bob, 30, and Ashley, 26, are an average white couple. How many years can Ashley expect to live as a
widow? 7.9 years
c) Why do the life expectancy curves intersect in the
accompanying figure?
At 80 both males and females can expect about 5 more
years.
90
150
100
Stocks
50
Bonds
0
0
5
10
15
Number of years after 1990
FIGURE FOR EXERCISE 92
Life expectancy
(years)
274
85
White
females
80
75
70
20
White
males
40
60
Present age
80
FIGURE FOR EXERCISES 95 AND 96
5.3
Addition, Subtraction, and Multiplication of Polynomials
GET TING MORE INVOLVED
97. Discussion. For which values of a and b is it true that
(ab)1 a1b1? Find a pair of nonzero values for a and
b for which (a b)1 a1 b1.
3
98. Writing. Explain how to evaluate 2 in three differ3
ent ways.
99. Discussion. Which of the following expressions has a
value different from the others? Explain.
a) 11
b) 30
c) 21 21
d) (1)2
e) (1)3 d
100. True or False? Explain your answer.
a) The square of a product is the product of the squares.
b) The square of a sum is the sum of the squares.
a) True
b) False
G R A P H I N G C ALC U L ATO R
EXERCISES
101. At 12% compounded annually the value of an investment
of $10,000 after x years is given by
y 10,000(1.12)x.
5.3
In this
section
●
Polynomials
●
Evaluating Polynomials
●
Addition and Subtraction
of Polynomials
●
Multiplication
of Polynomials
(5-19)
275
a) Graph y 10,000(1.12)x and the function y 20,000
on a graphing calculator. Use a viewing window that
shows the intersection of the two graphs.
b) Use the intersect feature of your calculator to find the
point of intersection.
c) The x-coordinate of the point of intersection is the number of years that it will take for the $10,000 investment
to double. What is that number of years?
b) (6.116, 20,000)
c) 6.116 years
102. The function y 72.2(1.002)x gives the life expectancy y
of a U.S. white male with present age x. (See Exercise 95.)
a) Graph y 72.2(1.002)x and y 86 on a graphing
calculator. Use a viewing window that shows the
intersection of the two graphs.
b) Use the intersect feature of your calculator to find the
point of intersection.
c) What does the x-coordinate of the point of intersection
tell you?
b) (87.54, 86)
c) At 87.54 years of age you can expect to live until
86. The model fails here.
ADDITION, SUBTRACTION, AND
MULTIPLICATION OF POLYNOMIALS
A polynomial is a particular type of algebraic expression that serves as a fundamental building block in algebra. We used polynomials in Chapters 1 and 2, but we
did not identify them as polynomials. In this section you will learn to recognize
polynomials and to add, subtract, and multiply them.
Polynomials
The expression 3x 3 15x 2 7x 2 is an example of a polynomial in one
variable. Because this expression could be written as
3x 3 (15x 2) 7x (2),
we say that this polynomial is a sum of four terms:
3x 3,
15x 2, 7x,
and
2.
A term of a polynomial is a single number or the product of a number and one or
more variables raised to whole number powers. The number preceding the variable
in each term is called the coefficient of that variable. In 3x 3 15x 2 7x 2 the
coefficient of x 3 is 3, the coefficient of x 2 is 15, and the coefficient of x is 7. In
algebra a number is frequently referred to as a constant, and so the last term 2 is
called the constant term. A polynomial is defined as a single term or a sum of a
finite number of terms.