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5 Exponents, Order of Operations, and Properties of Real Numbers

5 Exponents, Order of Operations, and Properties of Real Numbers

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Section 1.5



Exponents, Order of Operations, and Properties of Real Numbers



47



Keep in mind that an exponent applies only to the factor (number) directly

preceding it. Parentheses are needed to include a negative sign or other factors as

part of the base.

EXAMPLE 1

a. 25 ϭ 2 и 2



Evaluating Exponential Expressions



и2и2и2



ϭ 32

b.



΂΃

2

3



4



Rewrite expression as a product.

Simplify.



2

ϭ

3



2

3



2

3



и и и



2

3



и2и2и2

и3и3и3



ϭ



2

3



ϭ



16

81



Rewrite expression as a product.



Multiply fractions.



Simplify.



CHECKPOINT Now try Exercise 17.



EXAMPLE 2



Evaluating Exponential Expressions



a. ͑Ϫ4͒3 ϭ ͑Ϫ4͒͑Ϫ4͒͑Ϫ4͒

ϭ Ϫ64



Simplify.



b. ͑Ϫ3͒ ϭ ͑Ϫ3͒͑Ϫ3͒͑Ϫ3͒͑Ϫ3͒

ϭ 81

4



c. Ϫ34 ϭ Ϫ ͑3 и 3

ϭ Ϫ81



Rewrite expression as a product.



и 3 и 3͒



Rewrite expression as a product.

Simplify.

Rewrite expression as a product.

Simplify.



CHECKPOINT Now try Exercise 23.



In parts (a) and (b) of Example 2, note that when a negative number is raised

to an odd power, the result is negative, and when a negative number is raised to

an even power, the result is positive.

EXAMPLE 3



Transporting Capacity



A truck can transport a load of motor oil that is 6 cases high, 6 cases wide, and

6 cases long. Each case contains 6 quarts of motor oil. How many quarts can the

truck transport?

Premium



Motor

Oil

1 Quart



r

Moto

Oil

Premium



Motor

Oil

1 Quart



r

Moto

Oil

Premium



Motor

Oil

1 Quart



r

Moto

Oil



r

Moto

Oil

Premium



Motor

Oil

1 Quart



r

Moto

Oil

Premium



Motor

Oil

1 Quart



r

Moto

Oil

Premium



Motor

Oil

1 Quart



r

Moto

Oil



Premium



Motor

Oil

1 Quart



r

Moto

Oil



Premium



Motor

Oil

1 Quart



r

Moto

Oil



Premium



Motor

Oil

1 Quart



r

Moto

Oil



Premium



Motor

Oil

1 Quart



r

Moto

Oil



Premium



Motor

Oil

1 Quart



r

Moto

Oil



Premium



Motor

Oil

1 Quart



6

Figure 1.33



Premium



Motor

Oil

1 Quart



r

Moto

Oil

Premium



Motor

Oil

1 Quart



r

Moto

Oil

Premium



Motor

Oil

1 Quart



r

Moto

Oil



6



r

Moto

Oil

Premium



Motor

Oil

1 Quart



r

Moto

Oil

Premium



Motor

Oil



Solution

A sketch can help you solve this problem. From Figure 1.33, there are 6 и 6 и 6

cases of motor oil, and each case contains 6 quarts. You can see that 6 occurs as

a factor four times, which implies that the total number of quarts is



1 Quart



r

Moto

Oil

Premium



Motor

Oil



6



͑6 и 6 и 6͒ и 6 ϭ 64 ϭ 1296.



1 Quart



So, the truck can transport 1296 quarts of oil.

CHECKPOINT Now try Exercise 125.



48



Chapter 1



The Real Number System



2



᭤ Evaluate expressions using order of

operations.



Order of Operations

Up to this point in the text, you have studied five operations of arithmetic—

addition, subtraction, multiplication, division, and exponentiation (repeated

multiplication). When you use more than one operation in a given problem, you

face the question of which operation to perform first. For example, without

further guidelines, you could evaluate 4 ϩ 3 и 5 in two ways.



Technology: Discovery

To discover if your calculator

performs the established order of

operations, evaluate 7 ϩ 5 и

3 Ϫ 24 Ϭ 4 exactly as it appears.

Does your calculator display 5 or

18? If your calculator performs the

established order of operations, it

will display 18.



Add First



Multiply First



?

4ϩ3и5ϭ7и5



?

4 ϩ 3 и 5 ϭ 4 ϩ 15



ϭ 35



ϭ 19



According to the established order of operations, the second evaluation is

correct. The reason for this is that multiplication has a higher priority than

addition. The accepted priorities for order of operations are summarized below.



Order of Operations

1. Perform operations inside symbols of grouping—͑ ͒ or ͓ ͔— or

absolute value symbols, starting with the innermost symbols.

2. Evaluate all exponential expressions.

3. Perform all multiplications and divisions from left to right.

4. Perform all additions and subtractions from left to right.



In the priorities for order of operations, note that the highest priority is given

to symbols of grouping such as parentheses or brackets. This means that when

you want to be sure that you are communicating an expression correctly, you can

insert symbols of grouping to specify which operations you intend to be

performed first. For instance, if you want to make sure that 4 ϩ 3 и 5 will be

evaluated correctly, you can write it as 4 ϩ ͑3 и 5͒.

EXAMPLE 4

Study Tip

When you use symbols of grouping

in an expression, you should

alternate between parentheses and

brackets. For instance, the

expression

10 Ϫ ͑3 Ϫ ͓4 Ϫ ͑5 ϩ 7͔͒͒

is easier to understand than

10 Ϫ ͑3 Ϫ ͑4 Ϫ ͑5 ϩ 7͒͒͒.



a. 7 Ϫ ͓͑5



Order of Operations



и 3͒ ϩ 23͔ ϭ 7 Ϫ ͓15 ϩ 23͔



ϭ 7 Ϫ ͓15 ϩ 8͔

ϭ 7 Ϫ 23

ϭ Ϫ16

2

b. 36 Ϭ ͑3 и 2͒ Ϫ 6 ϭ 36 Ϭ ͑9 и 2͒ Ϫ 6

ϭ 36 Ϭ 18 Ϫ 6

ϭ2Ϫ6

ϭ Ϫ4

CHECKPOINT Now try Exercise 45.



Multiply inside the parentheses.

Evaluate exponential expression.

Add inside the brackets.

Subtract.

Evaluate exponential expression.

Multiply inside the parentheses.

Divide.

Subtract.



Section 1.5



Exponents, Order of Operations, and Properties of Real Numbers



Order of Operations



EXAMPLE 5

a.



b.



΂ ΃΂13΃ ϭ 73 и 78 ϩ ΂Ϫ 53΃΂13΃

1

3

ϭ ϩ ΂Ϫ ΃

8

5



3 8

3

Ϭ ϩ Ϫ

7 7

5



΂



΃



49



΂

8 5

ϭ ΂ ΃

3 12



Invert divisor and multiply.



Multiply fractions.



ϭ



15 Ϫ8

ϩ

40

40



Find common denominator.



ϭ



7

40



Add fractions.



8 1 1

8 2

3

ϩ

ϭ

ϩ

3 6 4

3 12 12



΃



Find common denominator.



Add inside the parentheses.



ϭ



40

36



Multiply fractions.



ϭ



10

9



Simplify.



CHECKPOINT Now try Exercise 55.



EXAMPLE 6



Order of Operations



Evaluate the expression 6 ϩ



8ϩ7

Ϫ ͑Ϫ5͒.

32 Ϫ 4



Solution





8ϩ7

8ϩ7

Ϫ ͑Ϫ5͒ ϭ 6 ϩ

Ϫ ͑Ϫ5͒

2

3 Ϫ4

9Ϫ4



Evaluate exponential expression.



ϭ6ϩ



15

Ϫ ͑Ϫ5͒

9Ϫ4



Add in numerator.



ϭ6ϩ



15

Ϫ ͑Ϫ5͒

5



Subtract in denominator.



ϭ 6 ϩ 3 Ϫ ͑Ϫ5͒



Divide.



ϭ9ϩ5



Add.



ϭ 14



Add.



CHECKPOINT Now try Exercise 71.



In Example 6, note that a fraction bar acts as a symbol of grouping. For

instance,

8ϩ7

32 Ϫ 4



means ͑8 ϩ 7͒ Ϭ ͑32 Ϫ 4͒, not 8 ϩ 7 Ϭ 32 Ϫ 4.



50



Chapter 1



The Real Number System



3 ᭤ Identify and use the properties of

real numbers.



Properties of Real Numbers

You are now ready for the symbolic versions of the properties that are true about

operations with real numbers. These properties are referred to as properties of

real numbers. The table shows a verbal description and an illustrative example

for each property. Keep in mind that the letters a, b, and c represent real

numbers, even though only rational numbers have been used to this point.



Properties of Real Numbers: Let a, b, and c be real numbers.

Property

1. Commutative Property of Addition:

Two real numbers can be added in either order.



Example



aϩbϭbϩa

2. Commutative Property of Multiplication:

Two real numbers can be multiplied in either order.



3ϩ5ϭ5ϩ3



ab ϭ ba

3. Associative Property of Addition:

When three real numbers are added, it makes no difference

which two are added first.



4



4.



5.



6.



7.



8.



͑a ϩ b͒ ϩ c ϭ a ϩ ͑b ϩ c͒

Associative Property of Multiplication:

When three real numbers are multiplied, it makes no difference

which two are multiplied first.

͑ab͒c ϭ a͑bc͒

Distributive Property:

Multiplication distributes over addition.

a͑b ϩ c͒ ϭ ab ϩ ac

͑a ϩ b͒c ϭ ac ϩ bc

Additive Identity Property:

The sum of zero and a real number equals the number itself.

aϩ0ϭ0ϩaϭa

Multiplicative Identity Property:

The product of 1 and a real number equals the number itself.

aи1ϭ1иaϭa

Additive Inverse Property:

The sum of a real number and its opposite is zero.

a ϩ ͑Ϫa͒ ϭ 0



и ͑Ϫ7͒ ϭ Ϫ7 и 4



͑2 ϩ 6͒ ϩ 5 ϭ 2 ϩ ͑6 ϩ 5͒



͑3 и 5͒ и 2 ϭ 3 и ͑5 и 2͒

3͑8 ϩ 5͒ ϭ 3 и 8 ϩ 3 и 5

͑3 ϩ 8͒5 ϭ 3 и 5 ϩ 8 и 5

3ϩ0ϭ0ϩ3ϭ3



4



и1ϭ1и4ϭ4



3 ϩ ͑Ϫ3͒ ϭ 0



9. Multiplicative Inverse Property:

The product of a nonzero real number and its reciprocal is 1.

1

a и ϭ 1, a 0

a



8



1



и8ϭ1



Section 1.5



Exponents, Order of Operations, and Properties of Real Numbers



EXAMPLE 7



51



Identifying Properties of Real Numbers



Identify the property of real numbers illustrated by each statement.

a. 3͑6 ϩ 2͒ ϭ 3 и 6 ϩ 3 и 2

b. 5



1



и5ϭ1



c. 7 ϩ ͑5 ϩ 4͒ ϭ ͑7 ϩ 5͒ ϩ 4

d. ͑12 ϩ 3͒ ϩ 0 ϭ 12 ϩ 3

e. 4͑11͒ ϭ 11͑4͒

Solution

a. This statement illustrates the Distributive Property.

b. This statement illustrates the Multiplicative Inverse Property.

c. This statement illustrates the Associative Property of Addition.

d. This statement illustrates the Additive Identity Property.

e. This statement illustrates the Commutative Property of Multiplication.

CHECKPOINT Now try Exercise 79.



EXAMPLE 8



Using the Properties of Real Numbers



Complete each statement using the specified property of real numbers.

a. Commutative Property of Addition:

5ϩ9ϭ᭿

b. Associative Property of Multiplication:

6͑5



и 13͒ ϭ ᭿



c. Distributive Property:

4и3ϩ4и7ϭ᭿

Solution

a. By the Commutative Property of Addition, you can write

5 ϩ 9 ϭ 9 ϩ 5.

b. By the Associative Property of Multiplication, you can write

6͑5 и 13͒ ϭ ͑6 и 5͒13.

c. By the Distributive Property, you can write

4



и 3 ϩ 4 и 7 ϭ 4͑3 ϩ 7͒.



CHECKPOINT Now try Exercise 101.



One of the distinctive things about algebra is that its rules make sense. You

don’t have to accept them on “blind faith”—instead, you can learn the reasons

that the rules work. For instance, there are some basic differences among the

operations of addition, multiplication, subtraction, and division.



52



Chapter 1



The Real Number System



In the summary of properties of real numbers on page 50, all the properties

are listed in terms of addition and multiplication. The reason for this is that

subtraction and division lack many of the properties listed in the summary. For

instance, subtraction and division are not commutative. To see this, consider the

following.

7Ϫ5



5 Ϫ 7 and



12 Ϭ 4



4 Ϭ 12



Similarly, subtraction and division are not associative.

9 Ϫ ͑5 Ϫ 3͒

EXAMPLE 9



͑9 Ϫ 5͒ Ϫ 3 and 12 Ϭ ͑4 Ϭ 2͒



͑12 Ϭ 4͒ Ϭ 2



Geometry: Area



You measure the width of a billboard and find that it is 60 feet. You are told that

its height is 22 feet less than its width.

a. Write an expression for the area of the billboard.

b. Use the Distributive Property to rewrite the expression.

c. Find the area of the billboard.

Solution

a. Begin by drawing and labeling a diagram, as shown in Figure 1.34. To find an

expression for the area of the billboard, multiply the width by the height.

(60 − 22) ft



Area ϭ Width ϫ Height

ϭ 60͑60 Ϫ 22͒



60 ft



Figure 1.34



b. To rewrite the expression 60͑60 Ϫ 22͒ using the Distributive Property,

distribute 60 over the subtraction.

60͑60 Ϫ 22͒ ϭ 60͑60͒ Ϫ 60͑22͒

c. To find the area of the billboard, evaluate the expression in part (b) as

follows.

60͑60͒ Ϫ 60͑22͒ ϭ 3600 Ϫ 1320

ϭ 2280



Multiply.

Subtract.



So, the area of the billboard is 2280 square feet.

CHECKPOINT Now try Exercise 131.



From Example 9(b) you can see that the Distributive Property is also true for

subtraction. For instance, the “subtraction form” of a͑b ϩ c͒ ϭ ab ϩ ac is

a͑b Ϫ c͒ ϭ a͓b ϩ ͑Ϫc͔͒

ϭ ab ϩ a͑Ϫc͒

ϭ ab Ϫ ac.



Section 1.5



Exponents, Order of Operations, and Properties of Real Numbers



53



Concept Check

1. Consider the expression 35.

(a) What part of the exponential expression is the

number 3?

(b) What part of the exponential expression is the

number 5?

2. In your own words, describe the priorities for the

established order of operations.



3. In your own words, state the Associative Property

of Addition and the Associative Property of

Multiplication. Give an example of each.

4. In your own words, state the Commutative Property

of Addition and the Commutative Property of

Multiplication. Give an example of each.



Go to pages 58–59 to

record your assignments.



1.5 EXERCISES



Developing Skills

In Exercises 1– 8, rewrite in exponential form.

1.

2.

3.

4.

5.

6.

7.

8.



2и2и2и2и2

4и4и4и4и4и4

͑Ϫ5͒ и ͑Ϫ5͒ и ͑Ϫ5͒ и ͑Ϫ5͒



͑Ϫ3͒ и ͑Ϫ3͒ и ͑Ϫ3͒

͑Ϫ 14 ͒ и ͑Ϫ 14 ͒

͑Ϫ 35 ͒ и ͑Ϫ 35 ͒ и ͑Ϫ 35 ͒ и ͑Ϫ 35 ͒

Ϫ ͓͑1.6͒ и ͑1.6͒ и ͑1.6͒ и ͑1.6͒ и ͑1.6͔͒

Ϫ ͓͑8.7͒ и ͑8.7͒ и ͑8.7͔͒



In Exercises 9 –16, rewrite as a product.

9. ͑Ϫ3͒6

10. ͑Ϫ8͒2

11.

13.



͑38 ͒5

͑Ϫ 12 ͒7

͑Ϫ 45 ͒6



12.



͑113 ͒



4



14.

15. Ϫ ͑9.8͒3

16. Ϫ ͑0.01͒8

In Exercises 17–28, evaluate the expression. See

Examples 1 and 2.

17. 32

19. 26

1 3

21. ͑4 ͒

23. ͑Ϫ5͒3



18. 43

20. 53

4 4

22. ͑5 ͒

24. ͑Ϫ4͒2



26. Ϫ ͑Ϫ6͒3

28. ͑Ϫ1.5͒4



25. Ϫ42

27. ͑Ϫ1.2͒3



In Exercises 29–72, evaluate the expression. If it is not

possible, state the reason. Write fractional answers in

simplest form. See Examples 4, 5, and 6.

29.

31.

33.

34.

35.

37.

39.



4 Ϫ 6 ϩ 10

5 Ϫ ͑8 Ϫ 15͒



Խ



Խ



30. 8 ϩ 9 Ϫ 12

32. 13 Ϫ ͑12 Ϫ 3͒



17 Ϫ 2 Ϫ ͑6 ϩ 5͒

125 Ϫ 10 Ϫ ͑25 Ϫ 3͒

15 ϩ 3 и 4

36. 9 Ϫ 5 и 2

25 Ϫ 32 Ϭ 4

38. 16 ϩ 24 Ϭ 8

͑16 Ϫ 5͒ Ϭ ͑3 Ϫ 5͒

40. ͑19 Ϫ 4͒ Ϭ ͑7 Ϫ 2͒



Խ



Խ



͑10 Ϫ 16͒ и ͑20 Ϫ 26͒

͑14 Ϫ 17͒ и ͑13 Ϫ 19͒

͑45 Ϭ 10͒ и 2

͑38 Ϭ 5͒ и 4

͓360 Ϫ ͑8 ϩ 12͔͒ Ϭ 5

46. ͓127 Ϫ ͑13 ϩ 4͔͒ Ϭ 10

47. 5 ϩ ͑22 и 3͒

48. 181 Ϫ ͑13 и 32͒

2

2

49. ͑Ϫ6͒ Ϫ ͑48 Ϭ 4 ͒

50. ͑Ϫ3͒3 ϩ ͑12 Ϭ 22͒

41.

42.

43.

44.

45.



51. ͑3



и 59 ͒ ϩ 1 Ϫ 13



1

2

53. 18͑2 ϩ 3 ͒



52.

54.



͑ ͒ ϩ 2 Ϫ 32

2

4

4͑ Ϫ 3 ϩ 3 ͒

2 3

3 4



54

55.

57.

59.



Chapter 1



͑



7 7

1

25 16 Ϫ 8

28

7 2

3 3 Ϭ 15



͑͒



͒



56.

58.



3 ϩ ͓15 Ϭ ͑Ϫ3͔͒

16



1 Ϫ 32

Ϫ2

2

7 Ϫ 42

63.

0

61.



65.



62



0

ϩ1



52 ϩ 122

13

3и6Ϫ4и6

69.

5ϩ1



3

2

3

8



87. 12 и 9 Ϫ 12 и 3 ϭ 12͑9 Ϫ 3͒

88. ͑32 ϩ 8͒ ϩ 5 ϭ 32 ϩ ͑8 ϩ 5͒



͑23 ϩ 16 ͒

͑15 ͒ Ϭ 2532



60.



5 ϩ ͓͑Ϫ12͒ Ϭ 4͔

24



62.



22 ϩ 42

5



64.



0

32 Ϫ 12



66.



89.

90.

91.

92.



33 ϩ 1

0



96. ͓͑7 ϩ 8͒6]5 ϭ ͑7 ϩ 8͒͑6



68.



4ϩ6

ϩ5

22 ϩ 1



72. 11 Ϫ



΂



0.1

12



΃



24



33 Ϫ 30

ϩ1

8ϩ1



1.32 ϩ 4͑3.68͒

75.

1.5



΂



74. 1000 Ϭ 1 ϩ



97. Commutative Property of Addition:

18 ϩ 5 ϭ᭿

98. Commutative Property of Addition:



0.09

4



΃



8



4.19 Ϫ 7͑2.27͒

76.

14.8



In Exercises 77–96, identify the property of real

numbers illustrated by the statement. See Example 7.

77. 6͑Ϫ3͒ ϭ Ϫ3͑6͒

78. 16 ϩ 10 ϭ 10 ϩ 16

79. 5 ϩ 10 ϭ 10 ϩ 5

80. Ϫ2͑8͒ ϭ 8͑Ϫ2͒

81.

82.

83.

84.

85.



6͑3 ϩ 13͒ ϭ 6 и 3 ϩ 6 и 13

1и4ϭ4

Ϫ16 ϩ 16 ϭ 0

͑14 ϩ 2͒3 ϭ 14 и 3 ϩ 2 и 3

͑10 ϩ 3͒ ϩ 2 ϭ 10 ϩ ͑3 ϩ 2͒



86. 25 ϩ ͑Ϫ25͒ ϭ 0



и 5͒



In Exercises 97–108, complete the statement using the

specified property of real numbers. See Example 8.



In Exercises 73–76, use a calculator to evaluate the

expression. Round your answer to two decimal places.

73. 300 1 ϩ



1

7͑7 ͒ ϭ 1

Ϫ14 ϩ 0 ϭ Ϫ14

0 ϩ 15 ϭ 15

͑2 и 3͒4 ϭ 2͑3 и 4͒



93. 14͑3 ϩ 8͒ ϭ 14 ͑3͒ ϩ 14 ͑8͒

94. 7 и 12 ϩ 7 и 8 ϭ 7͑12 ϩ 8͒

95. 4͑3 и 10͒ ϭ ͑4 и 3͒10



82 Ϫ 23

4

5и3ϩ5и6

70.

7Ϫ2



67.



71. 7 Ϫ



The Real Number System



3 ϩ 12 ϭ᭿

99. Commutative Property of Multiplication:

10͑Ϫ3͒ ϭ᭿

100. Commutative Property of Multiplication:

5͑8 ϩ 3͒ ϭ᭿

101. Distributive Property:

6͑19 ϩ 2͒ ϭ᭿

102. Distributive Property:

5͑7 Ϫ 16͒ ϭ᭿

103. Distributive Property:

3 и 4 ϩ 5 и 4 ϭ᭿

104. Distributive Property:

105.

106.

107.

108.



͑4 Ϫ 9͒12 ϭ᭿

᭿

Associative Property of Addition:

18 ϩ ͑12 ϩ 9͒ ϭ᭿

Associative Property of Addition:

10 ϩ ͑8 ϩ 7͒ ϭ᭿

Associative Property of Multiplication:

12͑3 и 4͒ ϭ᭿

Associative Property of Multiplication:

͑4 и 11͒10 ϭ᭿



Section 1.5



Exponents, Order of Operations, and Properties of Real Numbers



In Exercises 109–116, find (a) the additive inverse and

(b) the multiplicative inverse of the quantity.

109.

111.

113.

115.



50



110. 12

112. Ϫ8

114. 34

116. 0.45



Ϫ1

Ϫ 12

0.2



122. 24 ϩ 39 ϩ ͑Ϫ24͒

ϭ 24 ϩ ͑Ϫ24͒ ϩ 39

ϭ 0 ϩ 39

ϭ 39

123.



In Exercises 117–120, simplify the expression using (a)

the Distributive Property and (b) order of operations.

117.

118.

119.

120.



͑79 ϩ 6͒ ϩ 29

ϭ 79 ϩ ͑6 ϩ 29 ͒

ϭ 79 ϩ ͑29 ϩ 6͒

7

2

ϭ ͑9 ϩ 9 ͒ ϩ 6

ϭ1ϩ6

ϭ7



3͑6 ϩ 10͒

4͑8 Ϫ 3͒

2

3 ͑9 ϩ 24͒

1

2 ͑4



55



124.



͑23 и 7͒ и 21

и ͑7 и 21͒

ϭ и ͑21 и 7͒

2

ϭ ͑3 и 21͒ и 7

ϭ 14 и 7



Ϫ 2͒



ϭ 23

2

3



In Exercises 121–124, justify each step.

121. 7 и 4 ϩ 9 ϩ 2 и 4



ϭ7и4ϩ2и4ϩ9

ϭ ͑7 и 4 ϩ 2 и 4͒ ϩ 9

ϭ ͑7 ϩ 2͒4 ϩ 9

ϭ9и4ϩ9

ϭ 9͑4 ϩ 1͒

ϭ 9͑5͒



ϭ 98



ϭ 45



Solving Problems

125. Capacity A truck can transport a load of propane

tanks that is 4 cases high, 4 cases wide, and 4 cases

long. Each case contains 4 propane tanks. How

many tanks can the truck transport?

126. Capacity A grocery store has a cereal display that

is 8 boxes high, 8 boxes wide, and 8 boxes long.

How many cereal boxes are in the display?

Geometry

of the region.



In Exercises 127 and 128, find the area



127.



3

3

6

3



3

9



128.



8



12

8

4

8



8



129. Sales Tax You purchase a sweater for $35.95.

There is a 6% sales tax, which means that the total

amount you must pay is 35.95 ϩ 0.06(35.95).

(a) Use the Distributive Property to rewrite the

expression.

(b) How much must you pay for the sweater

including sales tax?



56



Chapter 1



The Real Number System



130. Cost of a Truck A new truck can be paid for by

48 monthly payments of $665 each plus a down

payment of 2.5 times the amount of the monthly

payment. This means that the total amount paid for

the truck is 2.5͑665͒ ϩ 48͑665͒.

(a) Use the Distributive Property to rewrite the

expression.

(b) What is the total amount paid for the truck?



(36 − 9) in.



36 in.

Figure for 132



131.



Geometry The width of a movie screen is 30

feet and its height is 8 feet less than its width.



(30 − 8) ft



Geometry In Exercises 133 and 134, write an

expression for the perimeter of the triangle shown in the

figure. Use the properties of real numbers to simplify the

expression.

133.



2•6+3



30 ft



(a) Write an expression for the area of the movie

screen.

(b) Use the Distributive Property to rewrite the

expression.

(c) Find the area of the movie screen.

132.



3 + 11



8−2



Geometry A picture frame is 36 inches wide

and its height is 9 inches less than its width.

(a) Write an expression for the area of the picture

frame.

(b) Use the Distributive Property to rewrite the

expression.

(c) Find the area of the picture frame.



134.



8+2

2•3



6•2−4



Think About It In Exercises 135 and 136, determine

whether the order in which the two activities are

performed is “commutative.” That is, do you obtain the

same result regardless of which activity is performed

first?

135. (a)

(b)

136. (a)

(b)



“Put on your socks.”

“Put on your shoes.”

“Weed the flower beds.”

“Mow the lawn.”



Explaining Concepts

137.



Are Ϫ62 and ͑Ϫ6͒2 equal? Explain.



138.



Are 2



и 52 and 102 equal? Explain.



In Exercises 139–148, explain why the statement is

true. (The symbol means “is not equal to.”)

139. 4 и 62 242

140. Ϫ32 ͑Ϫ3͒͑Ϫ3͒



141. 4 Ϫ ͑6 Ϫ 2͒



4Ϫ6Ϫ2



8Ϫ6

4Ϫ6

2

143. 100 Ϭ 2 ϫ 50 1

142.



144.



16

2



и2



4



145. 5͑7 ϩ 3͒ 5͑7͒ ϩ 3

146. Ϫ7͑5 Ϫ 2͒ Ϫ7͑5͒ Ϫ 7͑2͒



Section 1.5



147.



8

0



Exponents, Order of Operations, and Properties of Real Numbers



153. Consider the rectangle shown in the figure.

(a) Find the area of the rectangle by adding the

areas of regions I and II.



0



148. 5͑15 ͒



0



149. Error Analysis

Ϫ9 ϩ



Describe and correct the error.



9 ϩ 20

9 20

Ϫ ͑Ϫ3͒ ϭ Ϫ9 ϩ ϩ

Ϫ ͑Ϫ3͒

3͑5͒

3

5



(b) Find the area of the rectangle by multiplying its

length by its width.

(c) Explain how the results of parts (a) and (b)

relate to the Distributive Property.



ϭ Ϫ9 ϩ 3 ϩ 4 Ϫ ͑Ϫ3͒

ϭ1



2



150. Error Analysis



Describe and correct the error.



7 Ϫ 3͑8 ϩ 1͒ Ϫ 15 ϭ 4͑8 ϩ 1͒ Ϫ 15

ϭ 4͑9͒ Ϫ 15



2



3



I



II



Geometry In Exercises 154 and 155, find the area

of the shaded rectangle in two ways. Explain how the

results are related to the Distributive Property.

154.



ϭ 36 Ϫ 15

ϭ 21



15



8



15 − 6



6



151. Match each expression in the first column with its

value in the second column.

Expression



57



Value

͑6 ϩ 2͒ и ͑5 ϩ 3͒ 19

͑6 ϩ 2͒ и 5 ϩ 3 22

64

6ϩ2и5ϩ3

6 ϩ 2 и ͑5 ϩ 3͒ 43

152. Using the established order of operations, which of

the following expressions has a value of 72? For

those that don’t, decide whether you can insert

parentheses into the expression so that its value is 72.

(a) 4 ϩ 23 Ϫ 7

(b) 4 ϩ 8 и 6

(c) 93 Ϫ 25 Ϫ 4

(d) 70 ϩ 10 Ϭ 5

(e) 60 ϩ 20 Ϭ 2 ϩ 32

(f) 35 и 2 ϩ 2



155.



11

3

7

7−3



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