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D. Properties of Real Numbers

# 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

Given that x is a real number,

xϩ0ϭx

Zero is the identity

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.

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

EXAMPLE 7

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

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.

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—

R–9

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.

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

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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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 ᮣ

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