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SECTION 1.3
87.
- 27.72
13.2
88.
- 126.7
36.2
Exponents, Roots, and Order of Operations
89.
- 100
- 0.01
90.
23
- 50
- 0.05
Solve each problem.
91. The highest temperature ever recorded in Juneau, Alaska, was 90°F. The lowest temperature
ever recorded there w as - 22°F. What is the dif ference between these tw o temperatures?
(Source: World Almanac and Book of Facts.)
92. On August 10, 1936, a temperature of 120
°F w as recorded in P onds, Arkansas. On
February 13, 1905, Ozark, Arkansas, recorded a temperature of - 29°F. What is the difference between these two temperatures? (Source: World Almanac and Book of Facts.)
93. Andrew McGinnis has $48.35 in his checking account. He uses his debit card to mak e purchases of $35.99 and $20.00, w hich overdraws his account. His bank charges his account
an overdraft fee of $28.50. He then deposits his paycheck for $66.27 from his part-time job
at Arby’s. What is the balance in his account?
94. Kayla Koolbeck has $37.50 in her checking account. She uses her debit card to mak e purchases of $25.99 and $19.34, w hich overdraws her account. Her bank char ges her account
an overdraft fee of $25.00. She then deposits her pa ycheck for $58.66 from her par t-time
job at Subway. What is the balance in her account?
1.3 Exponents, Roots, and Order of Operations
OBJECTIVES
1 Use exponents.
2 Find square roots.
3 Use the order of
operations.
Two or more numbers whose product is a third number are factors of that third number.
For example, 2 and 6 are f actors of 12, since 2 # 6 = 12. Other integer factors of 12
are 1, 3, 4, 12, - 1, - 2, - 3, - 4, - 6, and - 12.
OBJECTIVE 1
Use exponents. In algebra, w e use exponents as a w ay of writing
products of repeated factors. For example, the product 2 # 2 # 2 # 2 # 2 is written
2 # 2 # 2 # 2 # 2 = 2 5.
⎧
⎪
⎪
⎨
⎪
⎪
⎩
4 Evaluate algebraic
expressions for given
values of variables.
5 factors of 2
The number 5 shows that 2 is used as a f actor 5 times. The number 5 is the exponent,
and 2 is the base.
25
Exponent
Base
Read 2 5 as “2 to the f ifth power,” or “2 to the f ifth.” Multiplying the f ive 2s gives
2 5 = 2 # 2 # 2 # 2 # 2 = 32.
Exponential Expression
If a is a real number and n is a natural number, then
an ⴝ a a a . . . a,
#
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
# # #
n factors of a
where n is the exponent, a is the base, and a n is an exponential e xpression.
Exponents are also called powers.
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EXAMPLE
1
Using Exponential Notation
Write using exponents.
3
5
(b)
#
3
3 2
= a b
5
5
Read as
A 35 B
⎧
⎪
⎨
⎪
⎩
(a) 4 # 4 # 4
Here, 4 is used as a f actor 3 times.
4 # 4 # 4 = 43
3 factors of 4
2
2 factors of
3
5
“ 3 squared.”
5
Read 43 as “4 cubed.”
(c) (- 6)( - 6)(- 6)(- 6) = ( - 6) 4
Read (- 6) 4 as “ - 6 to the fourth power,” or “ - 6 to the fourth.”
3
(d) (0.3)(0.3)(0.3)(0.3)(0.3) = (0.3) 5
3
(e) x
#x#x#x#x#
NOW TRY
x = x6
Exercises 13, 15, 17, and 19.
(a) 3 # 3 = 3 squared, or 32
6
6
In parts (a) and (b) of Example 1, we used the terms squared and cubed to refer to
powers of 2 and 3, respecti vely. The term squared comes from the f igure of a square,
which has the same measure for both length and width, as shown in Figure 16(a). Similarly, the ter m cubed comes from the f igure of a cube. As shown in Figure 16(b), the
length, width, and height of a cube ha ve the same measure.
6
(b) 6 6 6 = 6 cubed, or 63
#
#
FI G URE 1 6
EXAMPLE
2
Evaluating Exponential Expressions
Evaluate.
(a) 52 = 5 # 5 = 25
52
5 is used as a factor 2 times.
= 5 # 5, NOT 5 # 2.
2 3
2
(b) a b =
3
3
#
2
3
#
2
8
=
3
27
2
3
is used as a factor 3 times.
(c) 2 6 = 2 # 2 # 2 # 2 # 2 # 2 = 64
NOW TRY
EXAMPLE
3
Evaluating Exponential Expressions with Negative Signs
Evaluate.
(a) (- 3) 5 = (- 3)( - 3)(- 3)(- 3)(- 3) = - 243
(b) (- 2) 6 = (- 2)( - 2)(- 2)(- 2)(- 2)( - 2) = 64
(c)
Exercises 21 and 27.
The base is - 3.
The base is - 2.
- 26
There are no parentheses. The exponent 6 applies only to the number 2, not to - 2.
- 2 6 = - (2 # 2 # 2 # 2 # 2 # 2) = - 64
The base is 2.
NOW TRY
Exercises 29, 31, and 33.
Examples 3(a) and (b) suggest the follo wing generalizations.
The product of an odd number of negative factors is negative.
The product of an even number of negative factors is positive.
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SECTION 1.3
Exponents, Roots, and Order of Operations
CAUTION As sho wn in Examples 3(b) and (c), it is impor
- an
between and
( - a) n.
- a n = - 1(a # a # a # . . . # a)
25
tant to distinguish
The base is a.
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
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n factors of a
(- a) n
= (- a)( - a) # . . . # (- a)
The base is - a.
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
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n factors of - a
Be careful when evaluating an exponential expression with a negative sign.
Find square roots. As we saw in Example 2(a), 52 = 5
5 = 25, so
5 squared is 25. The opposite (inverse) of squaring a number is called taking its square
root. For e xample, a square root of 25 is 5. Another square root of 25 is - 5, since
( - 5) 2 = 25. Thus, 25 has two square roots: 5 and - 5.
We write the positive or principal square root of a number with the symbol 1 ,
called a radical sign. For example, the positive or principal square root of 25 is writ125 = 5.
ten The
negative squar e r oot of 25 is written - 125 = - 5. Since the
square of any nonzero real number is positive, the square root of a negative number,
such as 1ⴚ25, is not a real number.
OBJECTIVE 2
EXAMPLE
4
#
Finding Square Roots
Find each square root that is a real number .
(a) 136 = 6, since 6 is positive and
(b) 10 = 0, since 0 2 = 0.
62 = 36.
(c)
9
3
3 2
9
= , since a b =
.
A 16 4
4
16
(d) 10.16 = 0.4, since (0.4) 2 = 0.16.
(e) 1100 = 10, since 10 2 = 100.
(f) - 1100 = - 10, since the negative sign is outside the radical sign.
(g) 1 - 100 is not a real number, because the negative sign is inside the radical sign.
No real number squared equals - 100.
Notice the difference among the square roots in par ts (e), (f ), and (g). Part (e) is
the positive or principal square root of 100, part (f ) is the negative square root of 100,
and part (g) is the square root of - 100, which is not a real number.
NOW TRY
Exercises 37, 41, 43, and 47.
CAUTION The symbol 1 is used onl y for the positive square root, e xcept that
10 = 0. The symbol - 1 is used for the negative square root.
3, what should
we do first—add 5 and 2 or multiply 2 and 3? When an expression involves more than
one operation symbol, we use the following order of operations.
OBJECTIVE 3
Use the order of operations. To simplify 5 + 2
#
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Order of Operations
1. Work separately above and below any fraction bar.
2. If grouping symbols such as parentheses ( ), brackets [ ], or absolute value
bars | | are present, start with the innermost set and work outward.
3. Evaluate all powers, roots, and absolute values.
4. Multiply or divide in order from left to right.
5. Add or subtract in order from left to right.
EXAMPLE
5
Using the Order of Operations
Simplify.
(a) 5 + 2 # 3
= 5 + 6
= 11
Multiply.
Add.
(b) 24 , 3 # 2 + 6
Multiplications and divisions are done in the order in which they appear from
left to right, so divide first.
24 ,
=
=
=
3 # 2 + 6
8 # 2 + 6
16 + 6
22
Divide.
Multiply.
Add.
NOW TRY
EXAMPLE
6
Exercises 53 and 57.
Using the Order of Operations
Simplify.
(a) 10 , 5 + 2 | 3 - 4 |
= 10 , 5 + 2 | - 1 |
= 10 , 5 + 2 # 1
= 2 + 2
= 4
Subtract inside the absolute value bars.
Take the absolute value.
Divide; multiply.
Add.
(b)
4 # 32 + 7 - (2 + 8)
= 4 # 32 + 7 - 10
Add inside parentheses.
#
= 4 9 + 7 - 10
Evaluate the power.
32 ⴝ 3 # 3,
NOT 3 # 2.
= 36 + 7 - 10
Multiply.
= 43 - 10
Add.
= 33
Subtract.
1 #
(c)
4 + (6 , 3 - 7)
2
1
= # 4 + (2 - 7)
Divide inside parentheses.
2
1
= # 4 + ( - 5)
Subtract inside parentheses.
2
= 2 + ( - 5)
Multiply.
= -3
Add.
NOW TRY
Exercises 65 and 71.
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SECTION 1.3
EXAMPLE
7
Exponents, Roots, and Order of Operations
27
Using the Order of Operations
5 + (- 2 3)(2)
.
6 # 19 - 9 # 2
Work separately above and below the fraction bar.
Simplify
5 + (- 2 3)(2)
6 # 19 - 9 # 2
5 + ( - 8)(2)
= #
6 3 - 9 # 2
5 - 16
=
18 - 18
- 11
=
0
Evaluate the power and the root.
Multiply.
Subtract.
Since division by 0 is undef ined, the given expression is undef ined.
NOW TRY
OBJECTIVE 4
Exercise 75.
Evaluate algebraic expressions for given values of variables. Any
sequence of numbers, v ariables, operation symbols, and/or g rouping symbols for med
in accordance with the r ules of algebra is called an algebraic expression.
6ab,
5m - 9n,
and
- 2(x 2 + 4y)
Algebraic expressions
Algebraic expressions have different numerical values for different values of the variables. We evaluate such expressions by substituting given values for the variables.
For example, if movie tickets cost $8 each, the amount in dollars you pay for x tickets can be represented by the algebraic expression 8x. We can substitute different numbers of tickets to get the costs of purchasing those tick ets.
EXAMPLE
8
Evaluating Algebraic Expressions
Evaluate each expression if m = - 4, n = 5, p = - 6, and q = 25.
Use parentheses
around substituted
values to avoid errors.
(a) 5m - 9n
= 5(- 4) - 9(5)
= - 20 - 45
= - 65
(b)
m + 2n
4p
- 4 + 2(5)
=
4(- 6)
- 4 + 10
=
- 24
6
1
=
= - 24
4
Substitute; let m = - 4 and n = 5.
Multiply.
Subtract.
Substitute; let m = - 4, n = 5, and p = - 6.
Work separately above and below the fraction bar.
Write in lowest terms; also,
a
-b
= - ba .
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(c) - 3m 3 - n 2(1q)
= - 3(- 4) 3 - (5) 2 (125)
Substitute; let m = - 4, n = 5, and q = 25.
= - 3(- 64) - 25(5)
= 192 - 125
= 67
Evaluate the powers and the root.
Multiply.
Subtract.
Notice the careful use of parentheses around substituted v alues.
NOW TRY
Exercises 79 and 85.
1.3 Exercises
NOW TRY
Exercise
True or False Decide whether each statement is true or false. If it is false , correct the statement so that it is true .
1. - 46 = ( - 4) 6
2. - 47 = ( - 4) 7
3. 116 is a positive number.
4. 3 + 5 # 6 = 3 + (5 # 6)
5. (- 2) 7 is a negative number.
6. ( - 2) 8 is a positive number.
7. The product of 8 positi ve factors and 8
negative factors is positive.
8. The product of 3 positi ve f actors and 3
negative factors is positive.
9. In the e xponential e xpression - 35, - 3
is the base.
10. 1a is positive for all positive numbers a.
Concept Check In Exercises 11 and 12, evaluate each exponential expression.
11. (a) 82
(c) ( - 8) 2
(b) - 82
(d) - (- 8) 2
12. (a) 43
(c) (- 4) 3
Write each expression by using exponents. See Example 1.
13. 10 # 10 # 10 # 10
1 1
16. #
2 2
19. z # z # z # z # z # z # z
14. 8 # 8 # 8
(b) - 43
(d) - ( - 4) 3
3 # 3 # 3 # 3 # 3
4 4 4 4 4
18. (- 4)(- 4)(- 4)(- 4)
15.
17. (- 9)(- 9)(- 9)
20. a # a # a # a # a
Evaluate each expression. See Examples 2 and 3.
21. 42
1 3
25. a b
5
29. (- 5) 3
33. - 36
22. 2 4
1 4
26. a b
6
30. (- 2) 5
34. - 46
23. 0.283
4 4
27. a b
5
31. (- 2) 8
35. - 84
24. 0.913
7 3
28. a b
10
32. (- 3) 6
36. - 10 3
Find each square root. If it is not a r eal number, say so. See Example 4.
37. 181
38. 164
39. 1169
41. - 1400
42. - 1900
43.
45. - 10.49
46. - 10.64
47. 1- 36
100
A 121
40. 1225
44.
225
A 169
48. 1- 121
49. Matching Match each square root with the appropriate v alue or description.
(a) 1144
A. - 12
(b) 1- 144
B. 12
(c) - 1144
C. Not a real number
50. Explain why 1- 900 is not a real number.
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Exponents, Roots, and Order of Operations
SECTION 1.3
29
Concept Check In Exercises 51 and 52, a represents a positive number.
51. Is - 1- a positive, negative, or not a real number?
52. Is - 1a positive, negative, or not a real number?
Simplify each expression. Use the order of operations. See Examples 5–7.
53. 12 + 3 # 4
54. 15 + 5 # 2
55. 6 # 3 - 12 , 4
56. 9 # 4 - 8 , 2
57. 10 + 30 , 2 # 3
58. 12 + 24 , 3 # 2
59. - 3(5) 2 - ( - 2)( - 8)
60. - 9(2) 2 - ( - 3)( - 2)
61. 5 - 7 # 3 - (- 2) 3
62. - 4 - 3 # 5 + 62
63. - 7( 136) - (- 2)(- 3)
64. - 8(164) - (- 3)(- 7)
65. 6 | 4 - 5 | - 24 , 3
66. - 4 | 2 - 4 | + 8 # 2
67. | - 6 - 5 | (- 8) + 32
68. (- 6 - 3) | - 2 - 3 | , 9
69. 6 +
2
5
(- 9) 3
8
2
71. - 14a - b , (2 # 6 - 10)
7
#
16
70. 7 -
(- 5 + 14)(- 2 2)
-5 - 1
74.
(- 9 + 116)(- 32)
-4 - 1
75.
2(- 5) + (- 3)( - 2)
- 8 + 32 - 1
76.
3( - 4) + (- 5)( - 8)
23 - 2 - 6
77.
-5 - 9
b - 6
-7
- 9 - 11 + 3 # 7
#
5
6
3
72. - 12a - b - (6 # 5 , 3)
4
73.
5 - 3a
3
(- 8) + 12
4
12 - (- 8)
b - 5(- 1 - 7)
3 # 2 + 4
- 9 - (- 7) - 3- 5 - (- 8)4
- 4a
78.
Evaluate each expression if a = - 3, b = 64, and c = 6. See Example 8.
79. 3a + 1b
80. - 2a - 1b
81. 1b + c - a 82. 1b - c + a
83. 4a 3 + 2c
84. - 3a 4 - 3c
85.
2c + a 3
4b + 6a
86.
3c + a 2
2b - 6c
Evaluate each expression if w = 4, x = - 34, y = 12, and z = 1.25. See Example 8.
87. wy - 8x
88. wz - 12y
89. xy + y 4
91. - w + 2x + 3y + z
92. w - 6x + 5y - 3z
93.
Solve each problem.
95. An appro ximation of the amount in billions of
dollars that Americans have spent on their pets
from 1998 to 2009 can be obtained b y substituting a given year for x in the expression
2.076x - 4125.
(Source: American P et Products Association.)
Approximate the amount spent in each y
ear.
Round answers to the nearest tenth.
(a) 1998
(b) 2005
(c) 2009
(d) How has the amount Americans have spent
on their pets changed from 1998 to 2009?
7x + 9y
w
90. xy - x 2
94.
7y - 5x
2w
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96. An approximation of federal spending on education in billions of dollars from 2001 through
2005 can be obtained using the e xpression
y = 9.0499x - 18,071.87,
where x represents the year. (Source: U.S. Department of the Treasury.)
(a) Use this expression to complete the table. Round answers to the nearest tenth.
Year
Education Spending
(in billions of dollars)
2001
37.0
2002
46.0
2003
2004
2005
(b) How has the amount of federal spending on education changed from 2001 to 2005?
1.4 Properties of Real Numbers
The study of an y object is simplif ied when we know the proper ties of the object. F or
example, a property of water is that it freezes when cooled to 0°C. Knowing this helps
us to predict the behavior of water.
The study of numbers is no different. The basic properties of real numbers studied
in this section reflect results that occur consistently in work with numbers, so they have
been generalized to apply to expressions with variables as well.
OBJECTIVES
1 Use the distributive
property.
2 Use the inverse
properties.
3 Use the identity
properties.
OBJECTIVE 1
4 Use the commutative
and associative
properties.
This idea is illustrated by the divided rectangle in Figure 17. Similarly,
and
so
5
2
2
Area of left part is 2 . 3 = 6.
Area of right part is 2 . 5 = 10.
Area of total rectangle is 2(3 + 5) = 16.
FI GU RE 1 7
2(3 + 5) = 2 # 8 = 16
2 # 3 + 2 # 5 = 6 + 10 = 16,
2(3 + 5) = 2 # 3 + 2 # 5.
and
so
5 Use the multiplication
property of 0.
3
Use the distributive property. Notice that
- 435 + (- 3)4 = - 4(2) = - 8
- 4(5) + ( - 4)(- 3) = - 20 + 12 = - 8,
- 435 + (- 3)4 = - 4(5) + (- 4)(- 3).
These examples are generalized to all real numbers as the distributive property
of multiplication with respect to addition, or simply the distributive property.
Distributive Property
For any real numbers a, b, and c,
a(b ⴙ c) ⴝ ab ⴙ ac
and
(b ⴙ c)a ⴝ ba ⴙ ca.
The distributive property can also be written
ab ⴙ ac ⴝ a(b ⴙ c)
and
ba ⴙ ca ⴝ (b ⴙ c)a