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 5313
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
6ϩ
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