MCGRAW-HILL HIGHER EDUCATION AND BLACKBOARD HAVE TEAMED UP
1.5 Exponential Expressions and the Order
In Sections 1.3 and 1.4, you learned how to perform operations with a pair of real numbers to obtain a third real number. In this section, you will learn to evaluate expressions involving several numbers and operations.
U 1 V Arithmetic Expressions
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
(32) 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:
(32) 55 525 3(2 5)31013
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.
Mid-Chapter Quiz Sections 1.1 through 1.4 Chapter 1
Graph each set of numbers on a number line.
1. The set of integers between 4 and 8
2. The real numbers between 4 and 8
3. The whole numbers less than or equal to 3
4. The real numbers less than or equal to 3
Perform the indicated operations.
5. 1 2 1
8 6. 5
6 3 4 7. 3
5 1
6 8. 2
3 8 9
9. 1933 10. 6(5)
11. 12(3) 12. 15 (3)
13. 12 (5) 14. 60 40
15. 11(6) 16. 56 (8) 17. 3
4 3
5 18. 3
7 5 6 19. 381
2
5 20. 0 34
Miscellaneous.
21. What is the distance between 0 and 5 on the number line?
22. Evaluate 8, 8, and 0. 23. Write 1
4as a decimal and a percent.
24. Convert 50 yards per minute to feet per second.
25. Convert 3
8to an equivalent fraction with a denominator of 32.
26. Reduce 2 3 4
6to lowest terms.
27. What is 44 0?
1-41 1.5 Exponential Expressions and the Order of Operations 41
E X A M P L E 1
UCalculator Close-Up V
One advantage of a graphing calcula- tor 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 denominator as shown here.
U 2 V Exponential Expressions
An arithmetic expression with repeated multiplication can be written by using exponents. For example,
2 2223 and 5 552.
The 3 in 23is the number of times that 2 occurs in the product 222, while the 2 in 52is the number of times that 5 occurs in 55. We read 23as “2 cubed” or “2 to the third power.” We read 52as “5 squared” or “5 to the second power.” In general, an expression of the form anis called an exponential expression and is defined as follows.
Exponential Expression For any counting number n,
anaaa. . .a.
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, 919.
E X A M P L E 2 Using exponential notation
Write each product as an exponential expression.
a) 6 6 6 6 6 b) (3)(3)(3)(3) c) 3
2 3 2 3
2
Solution
a) 6 6 6 6 665 b) (3)(3)(3)(3)(3)4 c) 3
2 3 2 3
2 323
Now do Exercises 13–20 Using grouping symbols
Evaluate each expression.
a) (36)(36) b) 3459 c) 4
5
(
9 8)
Solution
a) (3 6)(3 6)(3)(9) Evaluate within parentheses first.
27 Multiply.
b) 345914 Evaluate within absolute value symbols.
14 Find the absolute values.
3 Subtract.
c) 4
5
(
9 8)
12
4 Evaluate the numerator and denominator.
3 Divide.
Now do Exercises 1–12
dug84356_ch01b.qxd 9/14/10 9:23 AM Page 41
E X A M P L E 3 Writing an exponential expression as a product Write each exponential expression as a product without exponents.
a) y6 b) (2)4 c) 543 d) (0.1)2
Solution
a) y6y y y y y y b) (2)4(2)(2)(2)(2) c) 543 54 54 54
d) (0.1)2(0.1)(0.1)
Now do Exercises 21–26
To evaluate an exponential expression, write the base as many times as indicated by the exponent, and then multiply the factors from left to right.
Note that 339. We do not multiply the exponent and the base when evaluating an exponential expression.
Be especially careful with exponential expressions involving negative numbers.
An exponential expression with a negative base is written with parentheses around the base as in (2)4:
(2)4(2)(2)(2)(2)16 CAUTION
E X A M P L E 4 Evaluating exponential expressions Evaluate.
a) 33 b) (2)3 c) 234 d) (0.4)2
Solution
a) 333 3 39 327 b) (2)3(2)(2)(2)
4(2) 8 c) 234 23 23 23 23
4 9 2
3 2 3 2
8 7 2
3 1 8 6 1
d) (0.4)2(0.4)(0.4)0.16
Now do Exercises 27–42 UCalculator Close-Up V
You can use the power key for any power. Most calculators also have an x2key that gives the second power.
Note that parentheses must be used when raising a fraction to a power.
1-43 1.5 Exponential Expressions and the Order of Operations 43
To evaluate (24), use the base 2 as a factor four times, and then find the opposite:
(24) (2 2 2 2) (16) 16 We often omit the parentheses in (24)and simply write 24. So,
24 (24) 16.
To evaluate (2)4, use the base 2 as a factor four times, and then find the opposite:
(2)4 (16) 16
Be careful with 104and (10)4. It is tempting to evaluate these two the same. However, we have agreed that 104 (104), where the expo- nent is applied only to positive 10. The negative sign is handled last. So 104 10,000, a negative number. Likewise,12 1,22 4, and34 81.
U 3 V The Order of Operations
When we evaluate expressions, operations within grouping symbols are always performed first. For example,
(32) 5(5) 525 and (2 3)26236.
To make expressions look simpler, we often omit some or all parentheses. In this case, we must agree on the order in which to perform the operations. We agree to do multi- plication before addition and exponential expressions before multiplication. So,
32 53 10 13 and 2 322 918.
CAUTION
UHelpful Hint V
“Please Excuse My Dear Aunt Sally”
(PEMDAS) is often used as a memory aid for the order of oper- ations. Do Parentheses, Exponents, Multiplication and Division, then Addition and Subtraction. Multipli- cation and division have equal prior- ity. The same goes for addition and subtraction.
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) 104 (104) Rewrite using parentheses.
(10,000) Find 104.
10,000 Then find the opposite of 10,000.
c) (0.5)2 (0.5)(0.5) Use 0.5 as a factor two times.
(0.25) 0.25
d) (5 8)2 (3)2 Evaluate within parentheses first.
(9) Square 3 to get 9.
9 Take the opposite of 9 to get 9.
Now do Exercises 43–50
dug84356_ch01b.qxd 9/14/10 9:23 AM Page 43
We state the complete order of operations in the following box.
Order of Operations
1. Evaluate expressions within grouping symbols first. Parentheses and brackets are grouping symbols. Absolute value bars and fraction bars indicate group- ing and an operation.
2. Evaluate each exponential expression (in order from left to right).
3. Perform multiplication and division (in order from left to right).
4. Perform addition and subtraction (in order from left to right).
Multiplication and division have equal priority in the order of operations. If both appear in an expression, they are performed in order from left to right. The same holds for addition and subtraction. For example,
8 4 32 36 and 9356511.
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 6 Using the order of operations Evaluate each expression.
a) 2332 b) 2 53 442 c) 2 3 4 33 8
2
Solution
a) 23 328 9 Evaluate exponential expressions before multiplying.
72
b) 2 53 4422 53 416 Exponential expressions first 101216 Multiplication second
14 Addition and subtraction from left to right c) 2 3 433 8
2 2 3 427 8
2 Exponential expressions first 24274 Multiplication and division second 1 Addition and subtraction from left to right
Now do Exercises 51–66 UCalculator Close-Up V
Most calculators follow the same order of operations shown here.
Evaluate these expressions with your calculator.
E X A M P L E 7 Grouping symbols and the order of operations Evaluate.
a) 3 2(723) b) 3 73 4 c)
9 52
5 3
(
8 7)
Solution
a) 32(723)32(78) Evaluate within parentheses first.
32(1)
3(2) Multiply.
5 Subtract.
1-45 1.5 Exponential Expressions and the Order of Operations 45
b) 3 73 437 12 Evaluate within the absolute value symbols first.
35
35 Evaluate the absolute value.
2 Subtract.
c)
9 52
5 3
(
8
7) 25 12
21
12
4 3 Numerator and denominator are treated as if in parentheses.
Now do Exercises 67–80
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(79)]
b) 23(95)3
Solution
a) 64[5(79)]64[5(2)] Innermost parentheses first 64[7] Next evaluate within the brackets.
628 Multiply.
22 Subtract.
b) 23(9 5)3 23 43 Innermost grouping first 213 Evaluate within the first
absolute value.
2 13 Evaluate absolute values.
23 Multiply.
5 Subtract.
Now do Exercises 81–88 UCalculator Close-Up V
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. You might notice other grouping symbols on your cal- culator, but they may or may not be used for grouping. See your manual.
U 4 V Applications
E X A M P L E 9 Doubling your bet
A strategy among gamblers is to double your bet and bet again after a loss. The only prob- lem with this strategy is that you might run out of money before you get a win. A gambler loses $100 and employs this strategy. He keeps losing, six times in a row. His seventh bet will be 100 26dollars.
a) Find the amount of the seventh bet.
b) Find the total amount lost on the first six bets.
dug84356_ch01b.qxd 9/14/10 9:23 AM Page 45
Warm-Ups ▼
Fill in the blank.
1. An is the result of writing
numbers in a meaningful combination with the ordinary operations of arithmetic.
2. symbols indicate the order in which opera- tions are performed.
3. An expression is an expression of the form an.
4. The tells us the order in which to perform operations when grouping symbols are omitted.
True or false?
5. (3)26 6. (53)24 7. 53 24
8. ⏐56⏐⏐5⏐⏐6⏐ 9. 56 2(56)2 10. (23)22232 11. 5338 12. (53)38 13. 6
2
6 0
U1V Arithmetic Expressions Evaluate each expression. See Example 1.
1. (43)(59) 2.(57)(23) 3. 342 4 4.4935
5. 7 3
(
5
9) 6.
8 1
2 1
7. (6 5)(7) 8.6(5 7)
9. (3 7) 6 10.3(7 6) 11. 16 (82) 12.(168) 2
Exercises
UStudy Tips V
• Take notes in class. Write down everything that you can. As soon as possible after class, rewrite your notes. Fill in details and make corrections.
• If your instructor takes the time to work an example, it is a good bet that your instructor expects you to understand the concepts involved.
1.5
Solution
a) By the order of operations, 10026100646400. So the seventh bet is $6400.
b) Now find the total of the first six losses:
100 100 2100 22100 23100 24100 25 100200400 80016003200 6300
So the gambler has lost a total of $6300 on the first six bets.
Now do Exercises 121-124
1-47 1.5 Exponential Expressions and the Order of Operations 47
59. (3)3 23 60. 32 5(1)3 61. 21 36 32 62. 18 92 33 63. 3 23 5 22 64. 2 5 32 4 0 65.
2
8 23523 66. 426 1 3
2 33
Evaluate each expression. See Example 7.
67. (342)(6) 68. 3 (234) 5 69. (3 2 6)3 70. 5 2(3 2)3 71. 25(34 2) 72. (37)(46 2) 73. 325 6 74. 367 3 75. (325)3 28
76. 46 36 9 77. 3
7
4
1
0
6 78.
6 3
( (
8) 1
2
)
79. 80.
Evaluate each expression. See Example 8.
81. 34[96(25)]
82. 93[5(36)2]
83. 62[(23)210]
84. 3[(23)2(64)2]
85. 453(327) 86. 234(7262) 87. 23(73)9 88. 3(24)324
Evaluate each expression. Use a calculator to check.
89. 123 90. (12)3
91. (2)24(1)(3) 92. (2)24(2)(3) 93. 424(1)(3) 94. 324(2)(3) 95. (11)24(5)(0) 96. (12)24(3)(0) 97.523 42 98. 625(3)2 99.[32(4)]2 100. [62(3)]2 101.11 102. 417
322 ã 4 302 42 7932
973
U2V Exponential Expressions
Write each product as an exponential expression. See Example 2.
13. 4 4 4 4 14.1 1 1 1 1
15. (5)(5)(5)(5) 16.(7)(7)(7) 17. (y)(y)(y) 18.x x x x x 19. 3
7 3 7 3
7 3 7 3
7 20.
2 y
2 y
2 y
2 y
Write each exponential expression as a product without exponents. See Example 3.
21. 53 22.(8)4
23. b2 24.(a)5
25. 125
26. 11 3 23
Evaluate each exponential expression. See Examples 4 and 5.
27. 34 28. 53 29. 09
30. 012 31. (5)4 32. (2)5 33. (6)3 34. (12)2 35. (10)5 36. (10)6 37. (0.1)3 38. (0.2)2
39. 123 40. 233 41. 122
42. 232 43. 82 44. 72
45. 84 46. 74
47. (7 10)3 48. (6 9)4 49. (22) (32) 50. (34) (52)
U3V The Order of Operations Evaluate each expression. See Example 6.
51. 20 2 5 52. 30 6 5
53. 1165 54. 824
55. 32 22 56. 5 102
57. 3 24 6 58. 5 48 3
dug84356_ch01b.qxd 9/14/10 9:23 AM Page 47
103. 104.
105.3(1)25(1)4 106.2(1)25(1)6
107.52234 108. 5(2)232 109. 2962 110. 8 35421 111. 325[42(49)]
112.2[(34)35]7 113.1 55(91) 114.6377(52)
Use a calculator to evaluate each expression. Round approximate answers to four decimal places.
115.3.224(3.6)(2.2) 116.(4.5)24(2.8)(4.6) 117.(5.6)3[4.7(3.3)2]
118.9.83[1.2(4.49.6)2]
119.
120.
U4V Applications
Solve each problem. See Example 9.
121.Gambler’s ruin. A gambler bets $5 and loses. He doubles his bet and loses again. He continues this pattern, losing eight times in a row. His ninth bet will be 5 28dollars.
a) Calculate the amount of the ninth bet.
b) What is the total amount lost on the first eight bets?
122.Big profits. Big Bulldog Motorcycles showed a profit of
$50,000 in its first year of operation. The company plans to double the profit each year for the next 9 years.
a) What will be the amount of the profit in the tenth year?
b) What will be the total amount of profit for the first 10 years of business?
123.Population of the United States. In 2009 the population of the United States was 306.2 million (U.S. Census Bureau, www.census.gov). If the population continues to
4.563.22 3.44(6.26)
3.44(8.32) 6.895.43
3(7) 35 4(4)
22
grow at an annual rate of 1.05%, then the population in the year 2020 will be 306.2(1.0105)11million.
a) Evaluate the expression to find the predicted population in 2020 to the nearest tenth of a million people.
b) Use the accompanying graph to estimate the year in which the population will reach 400 million people.
Population (millions)
500 400 300
10
0 20 30 40
Years since 2009 200
Figure for Exercise 123
124. Population of Mexico. In 2009 the population of Mexico was 110.3 million. If Mexico’s population contin- ues to grow at an annual rate of 1.43%, then the population in 2020 will be 110.3(1.0143)11million.
a) Find the predicted population in 2020 to the nearest tenth of a million people.
b) Use the result of Exercise 123 to determine whether the United States or Mexico will have the greater increase in population between 2009 and 2020.
Getting More Involved 125. Discussion
How do the expressions (5)3,(53), 53, (5)3, and 1 53differ?
126. Discussion
How do the expressions (4)4, (44),44,(4)4, and1 44differ?
1-49 1.6 Algebraic Expressions 49