D. Properties of Real Numbers
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College Algebra G&M—
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R–6
CHAPTER R A Review of Basic Concepts and Skills
The Additive and Multiplicative Identities
Given that x is a real number,
xϩ0ϭx
Zero is the identity
for addition.
1#xϭx
One is the identity
for multiplication.
For any real number x, there is a real number Ϫx such that x ϩ 1Ϫx2 ϭ 0. The
number Ϫx is called the additive inverse of x, since their sum results in the additive identity. Similarly, the multiplicative inverse of any nonzero number x is 1x , since x # 1x ϭ 1
p q
(the multiplicative identity). This property can also be stated as q # p ϭ 1 1p, q 02 for
q
p
p
any rational number q. Note that q and p are reciprocals.
The Additive and Multiplicative Inverses
Given that p, q, and x represent real numbers 1p, q
02:
x ϩ 1Ϫx2 ϭ 0
p q
# ϭ1
q p
p
q
x and Ϫx are
additive inverses.
EXAMPLE 7
ᮣ
Determining Additive and Multiplicative Inverses
Replace the box to create a true statement:
# Ϫ3 x ϭ 1 # x
a.
b. x ϩ 4.7 ϩ
5
Solution
ᮣ
q
and p are
multiplicative inverses.
a.
b.
ϭx
5
5 # Ϫ3
, since
ϭ1
Ϫ3
Ϫ3 5
ϭ Ϫ4.7, since 4.7 ϩ 1Ϫ4.72 ϭ 0
ϭ
Now try Exercises 69 and 70
ᮣ
Note that if no coefficient is indicated, it is assumed to be 1, as in x ϭ 1x,
1x2 ϩ 3x2 ϭ 11x2 ϩ 3x2 , and Ϫ1x3 Ϫ 5x2 2 ϭ Ϫ11x3 Ϫ 5x2 2.
The distributive property of multiplication over addition is widely used in a
study of algebra, because it enables us to rewrite a product as an equivalent sum and
vice versa.
The Distributive Property of Multiplication over Addition
Given that a, b, and c represent real numbers:
a1b ϩ c2 ϭ ab ϩ ac
A factor outside a sum can be
distributed to each addend in
the sum.
ab ϩ ac ϭ a1b ϩ c2
A factor common to each addend
in a sum can be “undistributed”
and written outside a group.
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College Algebra G&M—
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7
Section R.1 Algebraic Expressions and the Properties of Real Numbers
EXAMPLE 8
ᮣ
Simplifying Expressions Using the Distributive Property
Apply the distributive property as appropriate. Simplify if possible.
a. 71p ϩ 5.22
Solution
ᮣ
WORTHY OF NOTE
From Example 8b we learn that a
negative sign outside a group
changes the sign of all terms within
the group: Ϫ12.5 Ϫ x2 ϭ Ϫ2.5 ϩ x.
b. Ϫ12.5 Ϫ x2
c. 7x3 Ϫ x3
d.
1
5
nϩ n
2
2
a. 71p ϩ 5.22 ϭ 7p ϩ 715.22
ϭ 7p ϩ 36.4
b. Ϫ12.5 Ϫ x2 ϭ Ϫ112.5 Ϫ x2
ϭ Ϫ112.52 Ϫ 1Ϫ121x2
ϭ Ϫ2.5 ϩ x
c. 7x3 Ϫ x3 ϭ 7x3 Ϫ 1x3
ϭ 17 Ϫ 12x3
ϭ 6x3
d.
1
5
1
5
n ϩ n ϭ a ϩ bn
2
2
2
2
6
ϭ a bn
2
ϭ 3n
D. You’ve just seen how
we can identify and use
properties of real numbers
Now try Exercises 71 through 78
ᮣ
E. Simplifying Algebraic Expressions
Two terms are like terms only if they have the same variable factors (the coefficient is
not used to identify like terms). For instance, 3x2 and Ϫ17x2 are like terms, while 5x3
and 5x2 are not. We simplify expressions by combining like terms using the distributive property, along with the commutative and associative properties. Many times the
distributive property is used to eliminate grouping symbols and combine like terms
within the same expression.
EXAMPLE 9
ᮣ
Simplifying an Algebraic Expression
Solution
ᮣ
712p2 ϩ 12 Ϫ 11p2 ϩ 32
Simplify the expression completely: 712p2 ϩ 12 Ϫ 1p2 ϩ 32 .
ϭ 14p ϩ 7 Ϫ 1p Ϫ 3
ϭ 114p2 Ϫ 1p2 2 ϩ 17 Ϫ 32
ϭ 114 Ϫ 12p2 ϩ 4
ϭ 13p2 ϩ 4
2
2
original expression; note coefficient of Ϫ1
distributive property
commutative and associative properties (collect like terms)
distributive property
result
Now try Exercises 79 through 88
ᮣ
The steps for simplifying an algebraic expression are summarized here:
To Simplify an Expression
1. Eliminate parentheses by applying the distributive property.
2. Use the commutative and associative properties to group like terms.
3. Use the distributive property to combine like terms.
E. You’ve just seen how
we can simplify algebraic
expressions
As you practice with these ideas, many of the steps will become more automatic.
At some point, the distributive property, the commutative and associative properties, as
well as the use of algebraic addition will all be performed mentally.
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College Algebra G&M—
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R–8
CHAPTER R A Review of Basic Concepts and Skills
R.1 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary.
ᮣ
1. A term consisting of a single number is called a(n)
term.
4. When 3 # 14 # 23 is written as 3 # 23 # 14, the
property has been used.
2. A term containing a variable is called a(n)
term.
5. Discuss/Explain why the additive inverse of Ϫ5 is
5, while the multiplicative inverse of Ϫ5 is Ϫ15.
3. The constant factor in a variable term is called the
.
6. Discuss/Explain how we can rewrite the sum
3x ϩ 6y as a product, and the product 21x ϩ 72 as a
sum.
DEVELOPING YOUR SKILLS
Identify the number of terms in each expression and the
coefficient of each term.
7. 3x Ϫ 5y
9. 2x ϩ
xϩ3
4
8. Ϫ2a Ϫ 3b
10.
nϪ5
ϩ 7n
3
11. Ϫ2x2 ϩ x Ϫ 5
12. 3n2 ϩ n Ϫ 7
13. Ϫ1x ϩ 52
14. Ϫ1n Ϫ 32
Translate each phrase into an algebraic expression.
15. seven fewer than a number
16. a number decreased by six
17. the sum of a number and four
18. a number increased by nine
Create a mathematical model using descriptive
variables.
29. The length of the rectangle is three meters less than
twice the width.
30. The height of the triangle is six centimeters less
than three times the base.
31. The speed of the car was fifteen miles per hour
more than the speed of the bus.
32. It took Romulus three minutes more time than
Remus to finish the race.
33. Hovering altitude: The helicopter was hovering
150 ft above the top of the building. Express the
altitude of the helicopter in terms of the building’s
height.
19. the difference between a number and five is
squared
20. the sum of a number and two is cubed
21. thirteen less than twice a number
22. five less than double a number
23. a number squared plus the number doubled
24. a number cubed less the number tripled
25. five fewer than two-thirds of a number
26. fourteen more than one-half of a number
27. three times the sum of a number and five,
decreased by seven
28. five times the difference of a number and two,
increased by six
34. Stacks on a cruise liner: The smoke stacks of the
luxury liner cleared the bridge by 25 ft as it passed
beneath it. Express the height of the stacks in terms
of the bridge’s height.
35. Dimensions of a city park: The length of a
rectangular city park is 20 m more than twice its
width. Express the length of the park in terms of
the width.
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College Algebra G&M—
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Section R.1 Algebraic Expressions and the Properties of Real Numbers
36. Dimensions of a parking lot: In order to meet the
city code while using the available space, a
contractor planned to construct a parking lot with a
length that was 50 ft less than three times its width.
Express the length of the lot in terms of the width.
37. Cost of milk: In 2010, a gallon of milk cost two
and one-half times what it did in 1990. Express the
cost of a gallon of milk in 2010 in terms of the
1990 cost.
38. Cost of gas: In 2010, a gallon of gasoline cost two
and one-half times what it did in 1990. Express the
cost of a gallon of gas in 2010 in terms of the 1990
cost.
39. Pest control: In her pest control business, Judy
charges $50 per call plus $12.50 per gallon of
insecticide for the control of spiders and certain
insects. Express the total charge in terms of the
number of gallons of insecticide used.
40. Computer repairs: As his reputation and referral
business grew, Keith began to charge $75 per
service call plus an hourly rate of $50 for the repair
and maintenance of home computers. Express the
cost of a service call in terms of the number of
hours spent on the call.
Evaluate each algebraic expression given x ؍2 and
y ؍؊3.
41. 4x Ϫ 2y
42. 5x Ϫ 3y
43. Ϫ2x2 ϩ 3y2
44. Ϫ5x2 ϩ 4y2
45. 2y2 ϩ 5y Ϫ 3
46. 3x2 ϩ 2x Ϫ 5
47. Ϫ213y ϩ 12
48. Ϫ312y ϩ 52
49. 3x2y
50. 6xy2
51. 1Ϫ3x2 2 Ϫ 4xy Ϫ y2 52. 1Ϫ2x2 2 Ϫ 5xy Ϫ y2
53. 12x Ϫ 13y
55. 13x Ϫ 2y2
57.
54. 32x Ϫ 12y
2
Ϫ12y ϩ 5
Ϫ3x ϩ 1
59. 1Ϫ12y # 4
56. 12x Ϫ 3y2
58.
2
12x ϩ 1Ϫ32
Ϫ3y ϩ 1
60. 7 # 1Ϫ27y
Evaluate each expression for integers from ؊3 to 3
inclusive. Verify results using a graphing calculator.
What input(s) give an output of zero?
61. x2 Ϫ 3x Ϫ 4
62. x2 Ϫ 2x Ϫ 3
63. Ϫ311 Ϫ x2 Ϫ 6
64. 513 Ϫ x2 Ϫ 10
65. x3 Ϫ 6x ϩ 4
66. x3 ϩ 5x ϩ 18
9
Rewrite each expression using the given property and
simplify if possible.
67. Commutative property of addition
a. Ϫ5 ϩ 7
b. Ϫ2 ϩ n
c. Ϫ4.2 ϩ a ϩ 13.6
d. 7 ϩ x Ϫ 7
68. Associative property of multiplication
a. 2 # 13 # 62
b. 3 # 14 # b2
#
#
c. Ϫ1.5 16 a2
d. Ϫ6 # 1Ϫ56 # x2
Replace the box so that a true statement results.
69. a. x ϩ 1Ϫ3.22 ϩ
b. n Ϫ 56 ϩ
70. a.
b.
ϭx
ϭn
# 23x ϭ 1x
#
n
ϭ 1n
Ϫ3
Apply the distributive property and simplify if possible.
71. Ϫ51x Ϫ 2.62
72. Ϫ121v Ϫ 3.22
73. 23 1Ϫ15p ϩ 92
2
74. 56 1Ϫ15
q ϩ 242
75. 3a ϩ 1Ϫ5a2
76. 13m ϩ 1Ϫ5m2
77. 23x ϩ 34x
78.
5
12 y
Ϫ 38y
Simplify by removing all grouping symbols (as needed)
and combining like terms.
79. 31a2 ϩ 3a2 Ϫ 15a2 ϩ 7a2
80. 21b2 ϩ 5b2 Ϫ 16b2 ϩ 9b2
81. x2 Ϫ 13x Ϫ 5x2 2
82. n2 Ϫ 15n Ϫ 4n2 2
83. 13a ϩ 2b Ϫ 5c2 Ϫ 1a Ϫ b Ϫ 7c2
84. 1x Ϫ 4y ϩ 8z2 Ϫ 18x Ϫ 5y Ϫ 2z2
85. 35 15n Ϫ 42 ϩ 58 1n ϩ 162
86. 23 12x Ϫ 92 ϩ 34 1x ϩ 122
87. 13a2 Ϫ 5a ϩ 72 ϩ 212a2 Ϫ 4a Ϫ 62
88. 213m2 ϩ 2m Ϫ 72 Ϫ 1m2 Ϫ 5m ϩ 42
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College Algebra G&M—
10
ᮣ
WORKING WITH FORMULAS
89. Electrical resistance: R ؍
kL
d2
The electrical resistance in a wire depends on the
length and diameter of the wire. This resistance can
be modeled by the formula shown, where R is the
resistance in ohms, L is the length in feet, and d is
the diameter of the wire in inches. Find the
resistance if k ϭ 0.000025, d ϭ 0.015 in., and
L ϭ 90 ft
ᮣ
R–10
CHAPTER R A Review of Basic Concepts and Skills
k
V
If temperature remains constant, the pressure of a
gas held in a closed container is related to the
volume of gas by the formula shown, where P is
the pressure in pounds per square inch, V is the
volume of gas in cubic inches, and k is a constant
that depends on given conditions. Find the pressure
exerted by the gas if k ϭ 440,310 and
V ϭ 22,580 in3.
90. Volume and pressure: P ؍
APPLICATIONS
Translate each key phrase into an algebraic expression, then evaluate as indicated.
91. Cruising speed: A turbo-prop airliner has a cruising
speed that is one-half the cruising speed of a 767 jet
aircraft. (a) Express the speed of the turbo-prop in
terms of the speed of the jet, and (b) determine the
speed of the airliner if the cruising speed of the jet is
550 mph.
92. Softball toss: Macklyn can throw a softball twothirds as far as her father. (a) Express the distance
that Macklyn can throw a softball in terms of the
distance her father can throw. (b) If her father can
throw the ball 210 ft, how far can Macklyn throw
the ball?
93. Dimensions of a lawn: The length of a rectangular
lawn is 3 ft more than twice its width. (a) Express the
length of the lawn in terms of the width. (b) If the
width is 52 ft, what is the length?
94. Pitch of a roof: To obtain the proper pitch, the
crossbeam for a roof truss must be 2 ft less than
three-halves the rafter. (a) Express the length of
the crossbeam in terms of the rafter. (b) If the rafter
is 18 ft, how long is the crossbeam?
ᮣ
95. Postage costs: In 2009, a first class stamp cost 29¢
more than it did in 1978. Express the cost of a
2009 stamp in terms of the 1978 cost. If a stamp
cost 15¢ in 1978, what was the cost in 2009?
96. Minimum wage: In 2009, the federal minimum
wage was $4.95 per hour more than it was in 1976.
Express the 2009 wage in terms of the 1976 wage.
If the hourly wage in 1976 was $2.30, what was it
in 2009?
97. Repair costs: The TV repair shop charges a flat fee
of $43.50 to come to your house and $25 per hour
for labor. Express the cost of repairing a TV in terms
of the time it takes to repair it. If the repair took
1.5 hr, what was the total cost?
98. Repair costs: At the local car dealership, shop
charges are $79.50 to diagnose the problem and $85
per shop hour for labor. Express the cost of a repair
in terms of the labor involved. If a repair takes 3.5 hr,
how much will it cost?
EXTENDING THE CONCEPT
99. If C must be a positive odd integer and D must be a
negative even integer, then C2 ϩ D2 must be a:
a. positive odd integer.
b. positive even integer.
c. negative odd integer.
d. negative even integer.
e. cannot be determined.
100. Historically, several attempts have been made to
create metric time using factors of 10, but our
current system won out. If 1 day was 10 metric
hours, 1 metric hour was 10 metric minutes, and
1 metric minute was 10 metric seconds, what time
would it really be if a metric clock read 4:3:5?
Assume that each new day starts at midnight.
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College Algebra Graphs & Models—
R.2
Exponents, Scientific Notation, and a Review of Polynomials
LEARNING OBJECTIVES
In Section R.2 you will review how to:
A. Apply properties of
A. The Properties of Exponents
exponents
B. Perform operations in
C.
D.
E.
An exponent is a superscript number or letter occurring to the upper right of a base number, and indicates how many times the base occurs as a factor. For b # b # b ϭ b3, we say b3
is written in exponential form. In some cases, we may refer to b3 as an exponential term.
Exponential Notation
For any positive integer n,
bn ϭ b # b # b # . . . # b
b # b # b # . . . # b ϭ bn
and
⎞
⎜
⎜
⎜
⎬
⎜
⎜
⎠
scientific notation
Identify and classify
polynomial expressions
Add and subtract
polynomials
Compute the product of
two polynomials
Compute special
products: binomial
conjugates and binomial
squares
⎞
⎜
⎜
⎜
⎬
⎜
⎜
⎠
F.
In this section, we review basic exponential properties and operations on polynomials.
Although there are five to eight exponential properties (depending on how you count
them), all can be traced back to the basic definition involving repeated multiplication.
n times
n times
The Product and Power Properties
There are two properties that follow immediately from this definition. When b3 is multiplied by b2, we have an uninterrupted string of five factors: b3 # b2 ϭ 1b # b # b2 # 1b # b2,
which can then be written as b5. This is an example of the product property of exponents.
Product Property of Exponents
WORTHY OF NOTE
In this statement of the product
property and the exponential
properties that follow, it is assumed
that for any expression of the form
0m, m 7 0 (hence 0m ϭ 0).
For any base b and positive integers m and n:
bm # bn ϭ bmϩn
In words, the property says, to multiply exponential terms with the same base, keep
the common base and add the exponents. A special application of the product property
uses repeated factors of the same exponential term, as in (x2)3. Using the product property, we have 1x2 2 1x2 21x2 2 ϭ x6. Notice the same result can be found more quickly by
#
multiplying the inner exponent by the outer exponent: 1x2 2 3 ϭ x2 3 ϭ x6. We generalize
this idea to state the power property of exponents. In words the property says, to raise
an exponential term to a power, keep the same base and multiply the exponents.
Power Property of Exponents
For any base b and positive integers m and n:
1bm 2 n ϭ bm n
#
EXAMPLE 1
ᮣ
Multiplying Terms Using Exponential Properties
Compute each product.
a. Ϫ4x3 # 12x2
b. 1p3 2 2 # 1p4 2 5
Solution
R–11
ᮣ
a. Ϫ4x3 # 12x2 ϭ 1Ϫ4 # 12 21x3 # x2 2
ϭ 1Ϫ22 1x3ϩ2 2
ϭ Ϫ2 x5
#
#
b. 1p3 2 2 # 1p4 2 5 ϭ p3 2 # p4 5
ϭ p6 # p20
ϭ p6ϩ20
ϭ p26
commutative and associative properties
simplify; product property
result
power property
simplify
product property
result
Now try Exercises 7 through 12 ᮣ
11
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College Algebra Graphs & Models—
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R–12
CHAPTER R A Review of Basic Concepts and Skills
The power property can easily be extended to include more than one factor within
the parentheses. This application of the power property is sometimes called the
product to a power property and can be extended to include any number of factors.
We can also raise a quotient of exponential terms to a power. The result is called the
quotient to a power property. In words the properties say, to raise a product or quotient
of exponential terms to a power, multiply every exponent inside the parentheses by the
exponent outside the parentheses.
Product to a Power Property
For any bases a and b, and positive integers m, n, and p:
1ambn 2 p ϭ amp # bnp
Quotient to a Power Property
For any bases a and b
0, and positive integers m, n, and p:
a
EXAMPLE 2
ᮣ
am p amp
b ϭ np
bn
b
Simplifying Terms Using the Power Properties
Simplify using the power property (if possible):
Ϫ5a3 2
a. 1Ϫ3a2 2
b. Ϫ3a2
c. a
b
2b
Solution
ᮣ
WORTHY OF NOTE
Regarding Examples 2a and 2b,
note the difference between the
expressions 1Ϫ3a2 2 ϭ 1Ϫ3 # a2 2 and
Ϫ3a2 ϭ Ϫ3 # a2. In the first, the
exponent acts on both the negative
3 and the a; in the second, the
exponent acts on only the a and
there is no “product to a power.”
a. 1Ϫ3a2 2 ϭ 1Ϫ32 2 # 1a1 2 2
ϭ 9a2
1Ϫ52 2 1a3 2 2
Ϫ5a3 2
b ϭ
c. a
2b
22b2
25a6
ϭ
4b2
b. Ϫ3a2 is in simplified form
Now try Exercises 13 through 24 ᮣ
Applications of exponents sometimes involve linking one exponential term with
another using a substitution. The result is then simplified using exponential properties.
EXAMPLE 3
ᮣ
Applying the Power Property after a Substitution
The formula for the volume of a cube is V ϭ S3, where S is
the length of one edge. If the length of each edge is 2x2:
a. Find a formula for volume in terms of x.
b. Find the volume if x ϭ 2.
Solution
ᮣ
a. V ϭ S
3
S
substitute 2x 2 for S
ϭ 12x2 2 3
ϭ 8x6
2x2
2x2
2x2
b. For V ϭ 8x ,
V ϭ 8122 6
substitute 2 for x
#
ϭ 8 64 or 512 122 6 ϭ 64
The volume of the cube would be 512 units3.
6
Now try Exercises 25 and 26 ᮣ