1.4
1.4
In this
section
●
Arithmetic Expressions
●
Exponential Expressions
●
Square Roots
●
Order of Operations
●
Algebraic Expressions
Evaluating Expressions
(1-23) 23
EVALUATING EXPRESSIONS
In algebra you will learn to work with variables. However, there is often nothing
more important than finding a numerical answer to a question. This section is concerned with computation.
Arithmetic Expressions
The result of writing numbers in a meaningful combination with the ordinary operations of arithmetic is called an arithmetic expression or simply an expression. An
expression that involves more than one operation is called a sum, difference, product, or quotient if the last operation to be performed is addition, subtraction, multiplication, or division, respectively. Parentheses are used as grouping symbols to
indicate which operations are performed first. The expression
5 (2 3)
is a sum because the parentheses indicate that the product of 2 and 3 is to be found
before the addition is performed. So we evaluate this expression as follows:
5 (2 3) 5 6 11
If we write (5 2)3, the expression is a product and it has a different value.
(5 2)3 7 3 21
Brackets [ ] are also used to indicate grouping. If an expression occurs within absolute value bars  , it is evaluated before the absolute value is found. So absolute
value bars also act as grouping symbols. We perform first the operations within the
innermost grouping symbols.
E X A M P L E
1
calculator
close-up
You can use parentheses to
control the order in which
your calculator performs the
operations in an expression.
Grouping symbols
Evaluate each expression.
a) 5[(2 3) 8]
b) 2[(4 5) 3 6 ]
Solution
a) 5[(2 3) 8] 5[6 8] Innermost grouping first.
5[2]
10
b) 2[(4 5) 3 6 ] 2[20 3 ]
2[20 3]
2[17]
34
■
Exponential Expressions
We use the notation of exponents to simplify the writing of repeated multiplication. The product 5 5 5 5 is written as 54. The number 4 in 54 is called
the exponent, and it indicates the number of times that the factor 5 occurs in the
product.
24
(1-24)
Chapter 1
The Real Numbers
Exponential Expression
For any natural number n and real number a,
a n a · a · a · . . . · a.
n factors of a
We call a the base, n the exponent, and a n an exponential expression.
We read a n as “the nth power of a” or “a to the nth power.” The exponential
expressions 35 and 106 are read as “3 to the fifth power” and “10 to the sixth power.”
We can also use the words “squared” and “cubed” for the second and third powers, respectively. For example, 52 and 23 are read as “5 squared” and “2 cubed,” respectively.
E X A M P L E
2
calculator
close-up
Powers are indicated on a
graphing calculator using a
caret (^). Most calculators also
have an x 2-key for squaring.
Note that parentheses are
necessary in (3)4. Without
parentheses, your calculator
should get 34 81. Try it.
Exponential expressions
Evaluate.
b) (3)4
a) 23
1
c) 2
5
Solution
a) 23 2 2 2 8
b) (3)4 (3)(3)(3)(3) 81
15
1
1
1
1
1
1
c) ■
2
2
2
2
2
2
32
Square Roots
Because 32 9 and (3)2 9, both 3 and 3 are square roots of 9. We use the
radical symbol to indicate the nonnegative or principal square root of 9. We
write 9 3.
Square Roots
If a b, then a is called a square root of b. If a 0, then a is called the
a.
principal square root of b and we write b
2
The radical symbol is a grouping symbol. We perform all operations within the
radical symbol before the square root is found.
E X A M P L E
3
Evaluating square roots
Evaluate.
4
b) 9
6
1
a) 6
c) 3
(1
7
)
5
Solution
a) Because 82 64, we have 6
4 8.
b) Because the radical symbol is a grouping symbol, add 9 and 16 before finding
the square root:
6
1 2
5 5
9
16 3 4 7. So 9
6
1 9
1
6.
Note that 9
(1
7
)
5 3
(1
2) 3
6 6
c) 3
■
1.4
Evaluating Expressions
(1-25) 25
Order of Operations
calculator
To simplify the writing of expressions, we often omit some grouping symbols. If we
saw the expression
close-up
523
Because the radical symbol
on most calculators cannot
be extended, parentheses are
used to group the expression
that is inside the radical.
written without parentheses, we would not know how to evaluate it unless we had
a rule for which operations to perform first. Expressions in which some or all
grouping symbols are omitted, are evaluated consistently by using a rule called the
order of operations.
Order of Operations
Evaluate inside any grouping symbols first. Where grouping symbols are
missing use the following order.
1. Evaluate each exponential expression (in order from left to right).
2. Perform multiplication and division (in order from left to right).
3. Perform addition and subtraction (in order from left to right).
“In order from left to right” means that we evaluate the operations in the order in
which they are written. For example,
20 3 6 60 6 10
and
10 3 6 7 6 13.
If an expression contains grouping symbols, we evaluate within the grouping symbols using the order of operations.
E X A M P L E
4
calculator
close-up
When parentheses are omitted, most (but not all), calculators follow the same order of
operations such as we do in
this text. Try these computations on your calculator. To
use a calculator effectively,
you must practice with it.
Order of operations
Evaluate each expression.
b) 9 23
a) 5 2 3
c) (6 42)2
d) 40 8 2 5 3
Solution
a) 5 2 3 5 6 Multiply first.
11
Then add.
3
b) 9 2 9 8 Evaluate the exponential expression first.
72
Then multiply.
c) (6 42)2 (6 16)2 Evaluate 42 within the parentheses first.
(10)2
Then subtract.
100
(10)(10) 100
d) Multiplication and division are done from left to right.
40 8 2 5 3 5 2 5 3
10 5 3
23
6
CAUTION
■
Don’t confuse 32 and (3)2. We interpret 32 as the oppo-
site of 32. So 32 (32) 9, whereas (3)2 (3)(3) 9.
26
(1-26)
Chapter 1
E X A M P L E
helpful
5
hint
“Everybody Loves My Dear
Aunt Sally” is often used as a
memory aid for the order of
operations. Do Exponents
and Logarithms, Multiplication and Division, and then
Addition and Subtraction.
Logarithms are discussed later
in this text.
The Real Numbers
The order of negative signs
Evaluate each expression.
b) 52
a) 24
c) (3 5)2
d) (52 4 7)2
Solution
a) To evaluate 24, find 24 first and then take the opposite. So 24 16.
b) 52 (52) 25
c) Evaluate within the parentheses first, square that result, then take the opposite.
(3 5)2 (2)2
4
2
2
d) (5 4 7) (25 28)2
(3)2
9
Evaluate within parentheses first.
Square 2 to get 4.
Evaluate 52 within the parentheses first.
Then subtract.
Square 3 to get 9, then take the opposite of
9 to get 9.
■
When an expression involves a fraction bar, the numerator and denominator are
each treated as if they are in parentheses. The next example illustrates how the fraction bar groups the numerator and denominator.
E X A M P L E
6
calculator
Order of operations in fractions
Evaluate each quotient.
10 8
a) 68
Solution
2
10 8
a) 2
68
close-up
1
Some calculators use the
built-up form for fractions 12 ,
()
but some do not (12). If your
calculator does not use the
built-up form, then you must
enclose numerators and denominators (that contain operations) in parentheses as
shown here.
62 2 7
b) 4 32
Evaluate the numerator and denominator separately.
Then divide.
6 2 7
36 14
b) 4 32
46
22
Evaluate the numerator and denominator separately.
2
11
Then divide.
2
■
Algebraic Expressions
The result of combining numbers and variables with the ordinary operations of
arithmetic (in some meaningful way) is called an algebraic expression. For
example,
x
2x 5y, 5x2, (x 3)(x 2), b2 4ac, 5, and 2
are algebraic expressions, or simply expressions. An expression such as 2x 5y
has no definite value unless we assign values to x and y. For example, if x 3 and
y 4, then the value of 2x 5y is found by replacing x with 3 and y with 4 and
evaluating:
2x 5y 2(3) 5(4) 6 20 14
1.4
M A T H
A T
Evaluating Expressions
(1-27) 27
W O R K
Nancy Gittins, Assistant Director of Financial Aid at Babson College, helps graduate
and undergraduate students to achieve their
goal of financing their educations. Because
recent tuition and fees can be as high as
$20,000 a year, many students need financial
FINANCIAL AID
aid to help defray these expenses. Federal
DIRECTOR
loans and state loans, as well as grants from
the federal and state levels, can be given to students who qualify. Ms. Gittins administers many of these loans and grants and helps students to understand the different options that are available. The interest rate for these loans is now a variable
rate that is tied to treasury bills. The rate can be as high as 9.0% and as low as 7.3%.
In Exercise 107 of this section you will work with interest rates compounded
annually.
Note the importance of the order of operations in evaluating an algebraic expression.
To find the value of the difference 2x 5y when x 2 and y 3, replace
x and y by 2 and 3, respectively, and then evaluate.
2x 5y 2(2) 5(3) 4 (15) 4 15 11
E X A M P L E
7
calculator
close-up
To evaluate a c 2, first store
the values for a and c using
the STO key. Then enter the
expression.
Value of an algebraic expression
Evaluate each expression for a 2, b 3, and c 4.
a) a c 2
b) a b2
c) b2 4ac
ab
d) cb
Solution
a) Replace a by 2 and c by 4 in the expression a c2.
a c 2 2 42 2 16 14
b) a b2 2 (3)2 2 9 7
c) b2 4ac (3)2 4(2)(4) 9 32 23
a b 2 (3) 5
d) cb
4 (3) 7
■
CAUTION
When you replace a variable by a negative number, be sure to
use parentheses around the negative number. If we were to omit the parentheses in
Example 7(c), we would get 32 4(2)(4) 41 instead of 23.
A symbol such as y1 is treated like any other variable. We read y1 as “y one” or
“y sub one.” The 1 is called a subscript. We can think of y1 as the “first y” and y2 as
the “second y.” We use the subscript notation in the following example.
(1-28)
Chapter 1
E X A M P L E
helpful
8
hint
Many of the expressions that
we evaluate in this section are
expressions that we will study
later in this text. We use the
expression in Example 8 to
find the slope of a line in
Chapter 3.
E X A M P L E
study
9
tip
If you don’t know how to get
started on the exercises, go
back to the examples. Cover
the solution in the text with a
piece of paper and see if you
can solve the example. After
mastering the examples, then
try the exercises again.
The Real Numbers
An algebraic expression with subscripts
y2 y1
Let y1 12, y2 5, x1 3, and x2 4. Find the value of .
x2 x1
Solution
Substitute the appropriate values into the expression
y2 y1 5 (12) 7
1.
x2 x1
4 (3)
7
■
When we evaluate an algebraic expression involving only one variable for many
values of that variable, we get a collection of data. A graph (picture) of these data
can give us useful information.
Reading a graph
The expression 0.85(220 A) gives the target heart rate for beneficial exercise for
an athlete who is A years old. Use the graph in Fig. 1.13 to estimate the target heart
rate for a 40-year-old athlete. Use the graph to estimate the age of an athlete with a
target heart rate of 170.
170
Target heart rate
28
160
150
140
130
0 10 15 20 25 30 35 40 45 50 55 60 65 70
Age
FIGURE 1.13
Solution
To find the target heart rate for a 40-year-old athlete, first draw a vertical line from
age 40 up to the graph as shown in Fig. 1.13. From the point of intersection, draw a
horizontal line to the heart rate scale. So the target heart rate for a 40-year-old athlete is about 153. To find the age corresponding to a heart rate of 170, first draw a
horizontal line from heart rate 170 to the graph as shown in the figure. From the
point of intersection, draw a vertical line down to the age scale. The heart rate of
■
170 corresponds to an age of about 20.
WARM-UPS
True or false? Explain your answer.
1. 23 6 False
3. 22 4 True
2. 1 22 4 True
4. 6 3 2 18 False
1.4
WARM-UPS
(continued)
5. (6 3) 2 81 False
7. 6 32 15 True
9. 3 (2) 5 False
1. 4
6. (6 3)2 18 False
8. (3)3 33 True
10. 7 8 7 8 False
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is an arithmetic expression?
An arithmetic expression is the result of writing numbers in
a meaningful combination with the ordinary operations of
arithmetic.
2. How do you know whether to call an expression a sum, a
difference, a product, or a quotient?
An expression is called a sum, a difference, a product, or a
quotient if the last operation to be performed is addition,
subtraction, multiplication, or division, respectively.
3. Why are grouping symbols used?
Grouping symbols are used to indicate the order in which
operations are to be performed.
4. What is an exponential expression?
An exponential expression is an expression of the form a n.
5. What is the purpose of the order of operations?
The order of operations tells us the order in which to perform operations when we omit grouping symbols.
6. What is the difference between 32 and (3)2?
The value of 32 is 9 and the value of (3)2 is 9.
Evaluate each expression. See Example 1.
7. (3 4) (2 5) 22
8. 3 2 2 6 1
9. 4[5 3 (2 5) ] 8
10. 2 (3 4) 6 36
11. (6 8)( 2 3 6) 14
12. 5(6 [(5 7) 4]) 0
Evaluate each radical. See Example 3.
20. 1
00
7
6
4
6
21. 3
10
)
9 8
23. 4(7
10
22. 2
5
9
4
24. (1
1
)(
218
)
5
Evaluate each expression. See Examples 4 and 5.
25. 4 6 2 8
26. 8 3 9 19
27. 5 6(3 5) 17
28. 8 3(4 6) 14
13
1 1 1 1
29. 3 2 4 2
1
24
30.
2 42 4
1
1 1
3
1
16
32 (8)2 3 58 32. 62 (3)3 63
(2 7)2 25
34. (1 3 2)3 125
2
3
5 2 200
36. 24 42 0
3
(5)(2) 40
38. (1)(2 8)3 216
(32 4)2 25
40. (6 23)4 16
60 10 3 2 5 6 7.5
1
42. 75 (5)(3) 6 540
2
43. 5.5 2.34 22.4841
31.
33.
35.
37.
39.
41.
44. 5.32 4 6.1
3.69
45. (1.3 0.31)(2.9 4.88)
46. (6.7 9.88)3
1.9602
32.157432
47. 388.8 (13.5)(9.6)
276.48
48. (4.3)(5.5) (3.2)(1.2)
8.86875
Evaluate each expression. See Example 6.
9 12
26
49. 2
50. 97
45
3 5
51. 6 (2)
Evaluate each exponential expression. See Example 2.
13. 25 32
14. 34 81
15. (1)4 1
1 6 1
1 2 1
16. (1)5 1
17. 18. 3
2
9
64
9
19. 4
(1-29) 29
Evaluating Expressions
3
1
14 (2)
52. 3 3
427
53. 6
329
32 (9)
55. 0
2 32
6 2(3)
54. 8 3(3)
24 5
56. 3
32 24
2
0
Evaluate each expression for a 1, b 3, and c 4.
See Example 7.
57. b2 4ac 7
58. a2
b
4c 7
bc 7
ab
4
59. 60. ba 4
ac
3
61. (a b )(a b ) 8 62. (a c )(a c ) 15
63. c2
c
2
1 5
64. b2 2b 3 0
a
2 b 1
5
c
c
65. 66. 3
a
c
c
2
a b b
67. a b 4
68. b c 1
(1-30)
Chapter 1
The Real Numbers
y2 y1
Find the value of for each choice of y1, y2, x1, and x 2.
x 2 x1
See Example 8.
10
69. y1 4, y2 6, x1 2, x 2 7 9
70. y1 3, y2 3, x1 4, x 2 5 0
3
71. y1 1, y2 2, x1 3, x 2 1 4
7
72. y1 2, y2 5, x1 2, x 2 6 4
73. y1 2.4, y2 5.6, x1 5.9, x 2 4.7 2.67
74. y1 5.7, y2 6.9, x1 3.5, x 2 4.2
18
Evaluate each expression without a calculator. Use a calculator
to check.
75. 22 5(3)2
76. 32 3(6)2
41
99
78. (3 3)62
77. (2 5)32
27
0
(1
4)(
6)
79. 52
1
81. [13 2(5)]2
9
4 (1)
83. 3 2
1
80. 62
(2
4)(
4)
2
82. [6 2(4)]2
4
2 (3)
84. 35
5
2
86. 3(1)2 5(1) 6
8
1
1 2
88. 8 6 1
2
2
85. 3(2)2 5(2) 4
26
1 2
1
87. 4 3 2
2
2
3
0
2
1
1
89. 6 2 90. 9 6 2
3
2
1
1 1 1 1
1 1 1 1
92. 91. 2 3 4 2
3 2 3 2
1
5
4
12
93. 6 3 7 7 5 94. 12 4 3 4 5 17
9
95. 3 7[4 (2 5)]
96. 9 2[3 (4 6)]
46
23
97. 3 4(2 4 6 )
98. 3 ( 4 5 )
3
4
99. 4[2 (5 3 )2]
100. [5 (3)]2 [4 (2)]2
8
100
How much larger is the target heart rate of a 25-year-old
woman than that of a 65-year-old woman? Use the accompanying graph to estimate the age at which a woman’s target heart rate is 115. 26 beats per minute, age 43
Target heart rate
30
160
150
140
130
120
110
100
90
M
Fe
ale
ma
le
0 10 20 30 40 50 60 70
Age
FIGURE FOR EXERCISES 101 AND 102
102. Male target heart rate. The algebraic expression
0.75(220 A) gives the target heart rate for beneficial
exercise for men, where A is the age of the man. Use the
algebraic expression to find the target heart rate for a
20-year-old and a 50-year-old man. Use the accompanying graph to estimate the age at which a man’s target heart
rate is 115. 150,127.5, age 67
Solve each problem.
103. Perimeter of a pool. The algebraic expression 2L 2W
gives the perimeter of a rectangle with length L and width
W. Find the perimeter of a rectangular swimming pool
that has length 34 feet and width 18 feet. 104 feet
104. Area of a lot. The algebraic expression for the area of a
trapezoid, 0.5h(b1 b2), gives the area of the property
shown in the figure. Find the area if h 150 feet, b1 260 feet, and b2 220 feet.
36,000 ft2 (square feet)
Solve each problem. See Example 9.
101. Female target heart rate. The algebraic expression
0.65(220 A) gives the target heart rate for beneficial
exercise for women, where A is the age of the woman.
220 ft
150 ft
260 ft
FIGURE FOR EXERCISE 104
105. Saving for retirement. The expression P(1 r)n gives
the amount of an investment of P dollars invested for n
years at interest rate r compounded annually. Long-term
corporate bonds have had an average yield of 6.2%
annually over the last 40 years (Fidelity Investments,
www.fidelity.com).
1.5
Amount (thousands of dollars)
Growth of a $10,000 investment
at 6.2% annual rate
120
(1-31) 31
P(1 r)n to find the actual amount of the debt at the time
the payments start. $5,500, $5,441.96
Amount of debt (dollars)
a) Use the accompanying graph to estimate the amount
of a $10,000 investment in corporate bonds after
30 years. $60,000
b) Use the given expression to calculate the value of a
$10,000 investment after 30 years of growth at 6.2%
compounded annually. $60,776.47
Properties of the Real Numbers
6,000
5,000
4,000
3,000
2,000
1,000
0
100
80
60
0
1
2
3
4
Time after making the loan (years)
FIGURE FOR EXERCISE 107
40
20
0
10
20
30
40
FIGURE FOR EXERCISE 105
106. Saving for college. The average cost of a B.A. in 2005
will be $252,000 at an Ivy League school (Fortune
Investors Guide). The principal that must be invested at
interest rate r compounded annually to have A dollars
n years in the future is given by the algebraic expression
A
.
(1 r)n
What investment in 1987 would amount to $252,000 in
2005 at 7% compounded annually? $74,557.71
107. Student loan. A college student borrowed $4,000 at 8%
compounded annually in her freshman year and did not
have to make payments until 4 years later. Use the accompanying graph to estimate the amount that she owes
at the time the payments start. Use the expression
1.5
In this
section
●
Commutative Properties
●
Associative Properties
●
Distributive Property
●
Identity Properties
●
Inverse Properties
●
Multiplication Property
of Zero
108. High cost of nursing care. The average cost for a oneyear stay in a nursing home in 1990 was $29,930 (Fortune
Investors Guide). In n years from 1990 the average cost
will be 29,930(1.05)n dollars. Find the projected cost for a
one-year stay in 2005. $62,222
109. Soaring cost of nursing care. Some economists project
that the average cost of a one-year stay in a nursing home
n years from 1990 will be 29,930(1.08)n dollars. How
much more would you pay for a one-year stay in 2005
using this expression rather than the expression in the last
exercise? $32,721
GET TING MORE INVOLVED
110. Discussion. Evaluate 5(5(5 3 6) 4) 7 and 3 53
6 52 4 5 7. Explain why these two expressions
must have the same value.
111. Cooperative learning. Find some examples of algebraic
expressions that are not mentioned in this text and explain
to your class what they are used for.
PROPERTIES OF THE REAL NUMBERS
You know that the price of a hamburger plus the price of a Coke is the same as the
price of a Coke plus the price of a hamburger. But, do you know which property of
the real numbers is at work in this situation? In arithmetic we may be unaware when
to use properties of the real numbers, but in algebra we need a better understanding
of those properties. In this section we will study the properties of the basic operations on the set of real numbers.
Commutative Properties
We get the same result whether we evaluate 3 7 or 7 3. With multiplication,
we have 4 5 5 4. These examples illustrate the commutative properties.