1-5 Exponential Expression and the Order of Operations (1-29) 29
1 1 1 3 1 17 GET TING MORE INVOLVED
83. 84.
5 6 30 5 4 20
95. Discussion. If you divide $0 among five people, how
3 2 1 1 much does each person get? If you divide $5 among zero
85. 86. 1 4
4 15 10 4 people, how much does each person get? What do these
questions illustrate?
Use a calculator to perform the indicated operation. 96. Discussion. What is the difference between the non-
Round answers to three decimal places. negative numbers and the positive numbers?
45.37 97. Writing. Why do we learn multiplication of signed num-
87. 7.562 88. (345) (28) 12.321
6 bers before division?
12.34 98. Writing. Try to rewrite the rules for multiplying and
89. (4.3)(4.5) 19.35 90. 4.113
3 dividing signed numbers without using the idea of absolute
0 value. Are your rewritten rules clearer than the original
91. 0 92. 0 (34.51) 0 rules?
6.345
23.44
93. 199.4 0 undefined 94. undefined
0
1.5 EXPONENTIAL EXPRESSIONS AND
THE ORDER OF OPERATIONS
In Sections 1.3 and 1.4 you learned how to perform operations with a pair of real
In this numbers to obtain a third real number. In this section you will learn to evaluate ex-
section pressions involving several numbers and operations.
● Arithmetic Expressions
● Exponential Expressions Arithmetic Expressions
● The Order of Operations The result of writing numbers in a meaningful combination with the ordinary oper-
ations of arithmetic is called an arithmetic expression or simply an expression.
Consider the expressions
(3 2) 5 and 3 (2 5).
The parentheses are used as grouping symbols and indicate which operation to per-
form first. Because of the parentheses, these expressions have different values:
(3 2) 5 5 5 25
3 (2 5) 3 10 13
Absolute value symbols and fraction bars are also used as grouping symbols. The
numerator and denominator of a fraction are treated as if each is in parentheses.
E X A M P L E 1 Using grouping symbols
Evaluate each expression.
a) (3 6)(3 6)
b) 3 4 5 9
4 (8)
c)
59
30 (1-30) Chapter 1 Real Numbers and Their Properties
Solution
a) (3 6)(3 6) (3)(9) Evaluate within parentheses first.
27 Multiply.
b) 3 4 5 9 1 4 Evaluate within absolute value symbols.
14 Find the absolute values.
3 Subtract.
4 (8) 12
c) Evaluate the numerator and denominator.
59 4
3 Divide. ■
calculator close-up
One advantage of a graphing calculator is that you can enter
an entire expression on its display and then evaluate it. If your
calculator does not allow built-up form for fractions, then you
must use parentheses around the numerator and denomina-
tor as shown here.
Exponential Expressions
An arithmetic expression with repeated multiplication can be written by using
study tip exponents. For example,
If you need help, do not hesi- 2 2 2 23 and 5 5 52.
tate to get it. Math has a way The 3 in 23 is the number of times that 2 occurs in the product 2 2 2, while the 2
of building upon the past.
in 52 is the number of times that 5 occurs in 5 5. We read 23 as “2 cubed” or “2 to
What you learn today will be
the third power.” We read 52 as “5 squared” or “5 to the second power.” In general,
used tomorrow, and what you
learn tomorrow will be used
an expression of the form an is called an exponential expression and is defined as
the day after. If you don’t follows.
straighten out problems im-
mediately, then you can get Exponential Expression
hopelessly lost. If you are For any counting number n,
having trouble, see your
instructor to find out what a n a a a . . . a.
help is available. n factors
We call a the base and n the exponent.
The expression an is read “a to the nth power.” If the exponent is 1, it is usually
omitted. For example, 91 9.
E X A M P L E 2 Using exponential notation
Write each product as an exponential expression.
3 3 3
a) 6 6 6 6 6 b) (3)(3)(3)(3) c)
2 2 2
1.5 Exponential Expressions and the Order of Operations (1-31) 31
Solution
a) 6 6 6 6 6 65
b) (3)(3)(3)(3) (3)4
3 3 3 3 3
c)
2 2 2 2 ■
E X A M P L E 3 Writing an exponential expression as a product
Write each exponential expression as a product without exponents.
5 3
a) y6 b) (2)4 c) d) (0.1)2
4
Solution
a) y6 y y y y y y
b) (2)4 (2)(2)(2)(2)
5 3 5 5 5
c)
4 4 4 4
d) (0.1) (0.1)(0.1)
2
■
To evaluate an exponential expression, write the base as many times as indicated
by the exponent, then multiply the factors from left to right.
E X A M P L E 4 Evaluating exponential expressions
Evaluate.
2 4
a) 33 b) (2)3 c) d) (0.4)2
3
Solution
a) 33 3 3 3 9 3 27
calculator
b) (2)3 (2)(2)(2)
4(2)
close-up 8
2 4 2 2 2 2
You can use the power key for c)
any power. Most calculators 3 3 3 3 3
also have an x 2 key that gives 4 2 2
the second power. Note that
9 3 3
parentheses must be used
when raising a fraction to a 8 2
power. 27 3
16
81
d) (0.4)2 (0.4)(0.4) 0.16
■
CAUTION Note that 33 9. We do not multiply the exponent and the
base when evaluating an exponential expression.
32 (1-32) Chapter 1 Real Numbers and Their Properties
Be especially careful with exponential expressions involving negative numbers.
study tip An exponential expression with a negative base is written with parentheses around
Find out what help is available the base as in (2)4:
at your school. Accompany-
ing this text are video tapes, (2)4 (2)(2)(2)(2) 16
solution manuals, and a com-
puter tutorial. Around most To evaluate (24), use the base 2 as a factor four times, then find the opposite:
campuses you will find tutors
available for hire, but most (24) (2 2 2 2) (16) 16
schools have a math lab
where you can get help for
We often omit the parentheses in (24) and simply write 24. So
free. Some schools even have
24 (24) 16.
free one-on-one tutoring
available through special To evaluate (2)4, use the base 2 as a factor four times, then find the opposite:
programs.
(2)4 (16) 16
E X A M P L E 5 Evaluating exponential expressions involving negative numbers
Evaluate.
a) (10)4 b) 104
c) (0.5) 2
d) (5 8)2
Solution
a) (10)4 (10)(10)(10)(10) Use 10 as a factor four times.
10,000
b) 10 (104)
4
Rewrite using parentheses.
(10,000) Find 104.
10,000 Then find the opposite of 10,000.
c) (0.5) (0.5)(0.5)
2
Use 0.5 as a factor two times.
(0.25)
0.25
d) (5 8) (3)2
2
Evaluate within parentheses first.
(9) Square 3 to get 9.
9 Take the opposite of 9 to get 9. ■
helpful hint The Order of Operations
“Please Excuse My Dear Aunt To simplify writing of expressions, parentheses are often omitted as in the expres-
Sally” is often used as a mem- sion 3 2 5. When no parentheses are present, we agree to perform multiplica-
ory aid for the order of opera- tion before addition. So
tions. Do Parentheses, Expo-
3 2 5 3 10 13.
nents, Multiplication and
Division, then Addition and To evaluate expressions consistently, we follow an accepted order of opera-
Subtraction. tions. When no grouping symbols are present, we perform operations in the follow-
ing order:
1.5 Exponential Expressions and the Order of Operations (1-33) 33
Order of Operations
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” does not mean to perform multiplication before division
or addition before subtraction. It means to perform them from left to right. For
example,
843236 and 9 3 5 6 5 11.
E X A M P L E 6 Using the order of operations
Evaluate each expression.
8
a) 23 32 b) 2 5 3 4 42 c) 2 3 4 33
2
calculator Solution
a) 23 32 8 9 Evaluate exponential expressions before multiplying.
close-up 72
b) 2 5 3 4 42 2 5 3 4 16 Exponential expressions first
Most calculators follow the
same order of operations 10 12 16 Multiplication second
shown here. Evaluate these ex- 14 Addition and subtraction from
pressions with your calculator. left to right
8 8
c) 2 3 4 33 2 3 4 27 Exponential expressions first
2 2
24 27 4 Multiplication and division second
1 Addition and subtraction from
left to right ■
When grouping symbols are used, we perform operations within grouping
symbols first. The order of operations is followed within the grouping symbols.
E X A M P L E 7 Grouping symbols and the order of operations
Evaluate.
958
a) 3 2(7 23) b) 3 7 3 4 c)
52 3(7)
Solution
a) 3 2(7 23) 3 2(7 8) Evaluate within parentheses first.
3 2(1)
3 (2) Multiply.
5 Subtract.
b) 3 7 3 4 3 7 12 Evaluate within the absolute value symbols first.
3 5
35 Evaluate the absolute value.
2 Subtract.
34 (1-34) Chapter 1 Real Numbers and Their Properties
958 12 12
c) 3 Numerator and denominator are
52 3(7) 25 21 4 treated as if in parentheses. ■
When grouping symbols occur within grouping symbols, we evaluate within the
innermost grouping symbols first and then work outward. In this case, brackets [ ]
can be used as grouping symbols along with parentheses to make the grouping
clear.
E X A M P L E 8 Grouping within grouping
Evaluate each expression.
a) 6 4[5 (7 9)] b) 2 3 (9 5) 3
Solution
a) 6 4[5 (7 9)] 6 4[5 (2)] Innermost parentheses first
6 4[7] Next evaluate within the brackets.
6 28 Multiply.
22 Subtract.
b) 2 3 (9 5) 3 2 3 4 3 Innermost grouping first
2 1 3 Evaluate within the first
absolute value.
2 1 3 Evaluate absolute values.
2 3 Multiply.
5 Subtract. ■
calculator close-up
Graphing calculators can handle grouping symbols within
grouping symbols. Since parentheses must occur in pairs,
you should have the same number of left parentheses as
right parentheses.
WARM-UPS
True or false? Explain your answer.
1. (3)2 6 False 2.5 3 2 4 False
3. (5 3)2 4 True 4. 5 6 5 6 False
5. 5 6 2 (5 6) 2 False 6.(2 3)2 22 32 False
7. 5 33 8 False 8.(5 3)3 8 True
6 0 66
9. 6 False 10. 0 True
2 2 2
1.5 Exponential Expressions and the Order of Operations (1-35) 35
1. 5 EXERCISES
Reading and Writing After reading this section write out the Evaluate each exponential expression. See Examples 4 and 5.
answers to these questions. Use complete sentences. 35. 34 81 36. 53 125 37. 09 0
1. What is an arithmetic expression? 38. 012
0 4
39. (5) 625 40. (2) 5 32
An arithmetic expression is the result of writing numbers in 41. (6) 216
3
42. (12) 144 43. (10)5 100,000
2
a meaningful combination with the ordinary operations of 6
44. (10) 45. (0.1)3 46. (0.2)2
arithmetic.
1,000,000 0.001 0.04
2. What is the purpose of grouping symbols?
1 3 1 2 3 8 1 2 1
The purpose of grouping symbols is to indicate the order in 47. 48. 49.
2 8 3 27 2 4
which to perform operations. 2 2 4
3. What is an exponential expression?
An exponential expression is an expression of the form an.
50.
3
9
51. 8 64
2
52. 7 49
2
53. (8)4 4096 54. (7)3 343
4. What is the difference between 36 and (3)6?
55. (7 10) 27 3
56. (6 9)4 81
The value of 36 is negative while the value of (3)6 is
positive. 57. (2 ) (3 ) 13
2 2
58. (34) (52) 56
5. What is the purpose of the order of operations? Evaluate each expression. See Example 6.
The order of operations tells us the order in which to per- 59. 32 22 36 60. 5 102 500
form operations when grouping symbols are omitted. 61. 3 2 4 6 18 62. 5 4 8 3 44
6. What were the different types of grouping symbols used in 63. (3)3 23 19 64. 32 5(1)3 14
this section? 65. 21 36 3 17 2
66. 18 92 33 21
Grouping symbols used in this section were parentheses, 67. 3 2 5 2 44
3 2
68. 2 5 32 4 0 1
absolute value bars, and the fraction bar. 8 12
Evaluate each expression. See Example 1. 69. 2 3 5 23 70. 4 2 6 33
2 3
7. (4 3)(5 9) 4 8. (5 7)(2 3) 10 18 25
9. 3 4 2 4 10. 4 9 3 5 Evaluate each expression. See Example 7.
1 13 71. (3 42)(6) 78 72. 3 (23 4) 5 180
7 (9) 8 2 73. (3 2 6) 0 3
74. 5 2(3 2)3 7
11. 8 12. 3
35 1 1 75. 2 5(3 4 2) 27 76. (3 7)(4 6 2) 32
13. (6 5)(7) 7 14. 6 (5 7) 29 77. 3 2 5 6 1 78. 3 6 7 3 12
15. (3 7) 6 16 16. 3 (7 6) 4 79. (32 5) 3 2 8 8
17. 16 (8 2) 4 18. (16 8) 2 1 80. 4 6 3 6 9 17
Write each product as an exponential expression. See 346 6 (8)2
81. 7 82. 29
Example 2. 7 10 3 (1)
19. 4 4 4 4 44 20. 1 1 1 1 1 15 7 9 32 32 2 · 4 1
83. 11 84. 2
21. (5)(5)(5)(5) (5) 4
22. (7)(7)(7) (7)3 973 30 2 4 2
23. (y)(y)(y) (y) 3
24. x x x x x x 5 Evaluate each expression. See Example 8.
85. 3 4[9 6(2 5)] 111
3 3 3 3 3 3 5 y y y y y 4
25. 26.
7 7 7 7 7 7 2 2 2 2 2 86. 9 3[5 (3 6)2] 3
Write each exponential expression as a product without expo- 87. 62 [(2 3)2 10] 21
nents. See Example 3. 88. 3[(2 3)2 (6 4)2] 15
27. 53 5 5 5 28. (8)4 (8)(8)(8)(8) 89. 4 5 3 (32 7) 1
29. b b b
2
30. (a)5 (a)(a)(a)(a)(a) 90. 2 3 4 (72 62) 29
91. 2 3 (7 3) 9 11
22222
1 5 1 1 1 1 1
31.
2 92. 3 (2 4)3 2 4 25
Evaluate each expression. Use a calculator to check.
32.
13 13
3 13 13
12 12 12 12 93. 1 23 9 94. (1 2)3 27
33. (0.22)4 (0.22)(0.22)(0.22)(0.22) 95. (2) 4(1)(3) 16
2
96. (2)2 4(2)(3) 20
34. (1.25)6 (1.25)(1.25)(1.25)(1.25)(1.25)(1.25) 97. 42 4(1)(3) 28 98. 32 4(2)(3) 33
99. (11) 4(5)(0) 121 100. (12)2 4(3)(0) 144
2
36 (1-36) Chapter 1 Real Numbers and Their Properties
101. 52 3 42 73 102. 62 5(3)2 81 Population of U.S.
Growth rate: 0.86%
103. [3 2(4)]2 25 104. [6 2(3)]2 144 350
105. 1 1 0 106. 4 1 7 2
Population (millions)
4 (4) 3 (7)
107. 2 108. 5
2 2 35 300
109. 3(1)2 5(1) 4 12
110. 2(1)2 5(1) 6 13
111. 5 22 34 82 112. 5 (2)2 32 0 250
113. 2 9 6 54
2 0 5 10 15
Years after 1998
20
114. 8 3 5 42 1 22
FIGURE FOR EXERCISE 125
115. 32 5[4 2(4 9)] 79
116. 2[(3 4)3 5] 7 19
117. 1 5 5 (9 1) 24 a) Find the predicted population in the year 2010 to the
nearest tenth of a million people. 123.3 million
118. 6 3 7 7 (5 2) 19
b) Will the U.S. or Mexico have the greater increase in
population between the years 1998 and 2010? (See the
Use a calculator to evaluate each expression. previous exercise.) U.S.
119. 3.22 4(3.6)(2.2) 41.92
120. (4.5)2 4(2.8)(4.6) 31.27 GET TING MORE INVOLVED
121. (5.63)3 [4.7 (3.3)2] 184.643547 127. Discussion. Which operation symbols are used also as
122. 9.83 [1.2 (4.4 9.6)2] 967.032 grouping symbols?
3.44 (8.32) The fraction bar is used as a grouping symbol and an
123. 8.0548 operation symbol. Also absolute value and square root act
6.89 5.43
as grouping symbols and indicate operations.
4.56 3.22
124. 0.8021 128. Discussion. What is wrong with each of the following
3.44 (6.26) expressions?
a) 6(4 (3 (5)) 5 (7
Solve each problem. b) (4 6)(5 3[6 4) 8] c)
3 2)
125. Population of the United States. In 1998 the population a) Too few closing parentheses.
of the United States was 270.1 million (U.S. Census b) Only a right parenthesis occurs inside the brackets.
Bureau, www.census.gov). If the population continues to c) Left parenthesis occurs in numerator and right paren-
grow at an annual rate of 0.86%, then the population in the thesis occurs in denominator.
year 2010 will be 270.1(1.0086)12 million. Find the pre- 129. Discussion. How do the expressions (5)3, (53), 53,
dicted population in 2010 to the nearest tenth of a million (5)3, and 1 53 differ?
people. 299.3 million (5)3 (53) 53 1 53 and (5)3 53
126. Population of Mexico. In 1998 the population of 130. Discussion. How do the expressions (4)4, (44), 44,
Mexico was 97.2 million (World Resources 1997–1998, (4)4, and 1 44 differ?
www.wri.org). If Mexico’s population continues to grow (44) 44 (4)4 1 · 44 and (4)4 44
at an annual rate of 2.0%, then the population in the year
2010 will be 97.2(1.02)12 million.
1.6 ALGEBRAIC EXPRESSIONS
In Section 1.5 you studied arithmetic expressions. In this section you will study
expressions that are more general—expressions that involve variables.
Identifying Algebraic Expressions
Since variables (or letters) are used to represent numbers, we can use variables in
arithmetic expressions. The result of combining numbers and variables with the
ordinary operations of arithmetic (in some meaningful way) is called an algebraic
