3: Real Numbers and Algebraic Expressions
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60
Chapter 2 • Real Numbers
1
ϭ 0.125
8
5
ϭ 0.3125
16
7
ϭ 0.28
25
2
ϭ 0.66666 . . .
3
1
ϭ 0.166666 . . .
6
1
ϭ 0.08333 . . .
12
14
ϭ 0.14141414 . . .
99
2
ϭ 0.4
5
The nonrepeating decimals are called “irrational numbers” and do appear in forms other
than decimal form. For example, 12, 13, and p are irrational numbers; a partial representation for each of these follows.
22 ϭ 1.414213562373 . . .
23 ϭ 1.73205080756887 . . . t
Nonrepeating decimals
p ϭ 3.14159265358979 . . .
(We will do more work with the irrational numbers in Chapter 9.)
The rational numbers together with the irrational numbers form the set of real numbers. The
following tree diagram of the real number system is helpful for summarizing some basic ideas.
Real numbers
Rational
Irrational
Ϫ
Integers
Ϫ
0
ϩ
Nonintegers
ϩ
Ϫ
ϩ
Any real number can be traced down through the diagram as follows.
5 is real, rational, an integer, and positive
Ϫ4 is real, rational, an integer, and negative
3
is real, rational, a noninteger, and positive
4
0.23 is real, rational, a noninteger, and positive
Ϫ0.161616 . . . is real, rational, a noninteger, and negative
17 is real, irrational, and positive
Ϫ12 is real, irrational, and negative
In Section 1.3, we associated the set of integers with evenly spaced points on a line
as indicated in Figure 2.4. This idea of associating numbers with points on a
−4 −3 −2 −1
0
1
2
3
4
Figure 2.4
line can be extended so that there is a one-to-one correspondence between points on a line
and the entire set of real numbers (as shown in Figure 2.5). That is to say, to each real
2.3 • Real Numbers and Algebraic Expressions
61
number there corresponds one and only one point on the line, and to each point on the line
there corresponds one and only one real number. The line is often referred to as the real
number line, and the number associated with each point on the line is called the coordinate
of the point.
−π
− 2
−1
2
−4 −3 −2 −1
1
2
0
π
2
1
2
3
4
Figure 2.5
The properties we discussed in Section 1.5 pertaining to integers are true for all real numbers; we restate them here for your convenience. The multiplicative inverse property was
added to the list; a discussion of that property follows.
Commutative Property of Addition
If a and b are real numbers, then
aϩbϭbϩa
Commutative Property of Multiplication
If a and b are real numbers, then
ab ϭ ba
Associative Property of Addition
If a, b, and c are real numbers, then
1a ϩ b2 ϩ c ϭ a ϩ 1b ϩ c2
Associative Property of Multiplication
If a, b, and c are real numbers, then
1ab2c ϭ a1bc2
Identity Property of Addition
If a is any real number, then
aϩ0ϭ0ϩaϭa
Identity Property of Multiplication
If a is any real number, then
a112 ϭ 11a2 ϭ a
62
Chapter 2 • Real Numbers
Additive Inverse Property
For every real number a, there exists a real number Ϫa, such that
a ϩ 1Ϫa2 ϭ 1Ϫa2 ϩ a ϭ 0
Multiplication Property of Zero
If a is any real number, then
a(0) ϭ 0(a) ϭ 0
Multiplicative Property of Negative One
If a is any real number, then
a1Ϫ12 ϭ Ϫ11a2 ϭ Ϫa
Multiplicative Inverse Property
For every nonzero real number a, there exists a real number
1
1
aa b ϭ (a) ϭ 1
a
a
1
, such that
a
Distributive Property
If a, b, and c are real numbers, then
a1b ϩ c2 ϭ ab ϩ ac
1
is called the multiplicative inverse or the reciprocal of a. For example,
a
1
1
1
1
1
the reciprocal of 2 is and 2 a b ϭ 122 ϭ 1. Likewise, the reciprocal of is ϭ 2.
2
2
2
2
1
2
1
Therefore, 2 and are said to be reciprocals (or multiplicative inverses) of each other.
2
The number
Also,
2
5
2
5
and
are multiplicative inverses, and a b a b ϭ 1. Since division by zero
5
2
5
2
is undefined, zero does not have a reciprocal.
Basic Operations with Decimals
The basic operations with decimals may be related to the corresponding operation with
3
4
7
common fractions. For example, 0.3 ϩ 0.4 ϭ 0.7 because
, and
ϩ
ϭ
10
10
10
2.3 • Real Numbers and Algebraic Expressions
63
37
24
13
Ϫ
ϭ
. In general, to add or subtract decimals, we
100 100 100
add or subtract the hundredths, the tenths, the ones, the tens, and so on. To keep place values
aligned, we line up the decimal points.
0.37 Ϫ 0.24 ϭ 0.13 because
Addition
1
2.14
3.12
5.16
10.42
Subtraction
1 11
5.214
3.162
7.218
8.914
24.508
616
7.6
4.9
2.7
81113
9.235
6.781
2.454
The following examples can be used to formulate a general rule for multiplying decimals.
Because
7
10
и 10 ϭ 100 , then (0.7)(0.3) ϭ 0.21
3
21
Because
9
10
#
Because
11
100
23
207
ϭ
, then (0.9)(0.23) ϭ 0.207
100
1000
#
13
143
ϭ
, then (0.11)(0.13) ϭ 0.0143
100
10,000
In general, to multiply decimals we (1) multiply the numbers and ignore the decimal points,
and then (2) insert the decimal point in the product so that the number of digits to the right
of the decimal point in the product is equal to the sum of the number of digits to the right of
the decimal point in each factor.
0.7
ϫ
0.3
ϭ
0.21
One digit
to right
ϩ
One digit
to right
ϭ
Two digits
to right
0.9
ϫ
0.23
ϭ
0.207
One digit
to right
ϩ
Two digits
to right
ϭ
Three digits
to right
0.11
ϫ
0.13
ϭ
0.0143
Two digits
to right
ϩ
Two digits
to right
ϭ
Four digits
to right
We frequently use the vertical format when multiplying decimals.
41.2
0.13
1236
412
5.356
One digit to right
Two digits to right
0.021
0.03
0.00063
Three digits to right
Two digits to right
Five digits to right
Three digits to right
Notice that in the last example we actually multiplied 3 и 21 and then inserted three 0s to the
left so that there would be five digits to the right of the decimal point.
64
Chapter 2 • Real Numbers
Once again let’s look at some links between common fractions and decimals.
3
0.3
6
6 # 1
3
Because
Ϭ2ϭ
ϭ , then 2ͤ 0.6
10
10 2
10
3
0.03
39
39 # 1
3
Because
Ϭ 13 ϭ
ϭ
, then 13ͤ 0.39
100
100 13
100
17
0.17
85
85 # 1
17
Because
Ϭ5ϭ
ϭ
, then 5ͤ 0.85
100
100 5
100
In general, to divide a decimal by a nonzero whole number we (1) place the decimal point in
the quotient directly above the decimal point in the dividend
Quotient
a Divisorͤ Dividend b
and then (2) divide as with whole numbers, except that in the division process,
zeros are placed in the quotient immediately to the right of the decimal point in
order to show the correct place value.
0.121
4ͤ 0.484
0.24
32ͤ 7.68
6 4
1 28
1 28
0.019
12ͤ 0.228
12
108
108
Zero needed to show the correct
place value
Don’t forget that division can be checked by multiplication. For example, since (12)(0.019) ϭ
0.228 we know that our last division example is correct.
Problems involving division by a decimal are easier to handle if we change the problem
to an equivalent problem that has a whole number divisor. Consider the following examples
in which the original division problem was changed to fractional form to show the reasoning
involved in the procedure.
0.4
0.24
0.24 10
2.4
0.6ͤ0.24 S
ϭa
ba b ϭ
S 6ͤ2.4
0.6
0.6
10
6
0.12ͤ0.156 S
1.3
0.156 100
15.6
0.156
ϭa
ba
bϭ
S 12ͤ 15.6
0.12
0.12
100
12
12 0
36
36
1.3ͤ0.026 S
0.02
0.026 10
0.26
0.026
ϭa
ba b ϭ
S 13ͤ0.26
1.3
1.3
10
13
0 26
The format commonly used with such problems is as follows.
5.6
The arrows indicate that the divisor and dividend were
x21.ͤ 1x17.6
multiplied by 100, which changes the divisor to a whole number
1 05
12 6
12 6
0.04
3x7.ͤ x1.48
The divisor and dividend were multiplied by 10
1 48
Our agreements for operating with positive and negative integers extend to all real numbers. For example, the product of two negative real numbers is a positive real number. Make
sure that you agree with the following results. (You may need to do some work on scratch
paper since the steps are not shown.)
0.24 ϩ (Ϫ0.18) ϭ 0.06
(Ϫ0.4)(0.8) ϭ Ϫ0.32
2.3 • Real Numbers and Algebraic Expressions
Ϫ7.2 ϩ 5.1 ϭ Ϫ2.1
(Ϫ0.5)(Ϫ0.13) ϭ 0.065
Ϫ0.6 ϩ (Ϫ0.8) ϭ Ϫ1.4
(1.4) Ϭ (Ϫ0.2) ϭ Ϫ7
2.4 Ϫ 6.1 ϭ Ϫ3.7
(Ϫ0.18) Ϭ (0.3) ϭ Ϫ0.6
0.31 Ϫ (Ϫ0.52) ϭ 0.83
(Ϫ0.24) Ϭ (Ϫ4) ϭ 0.06
65
(0.2)(Ϫ0.3) ϭ Ϫ0.06
Numerical and algebraic expressions may contain the decimal form as well as the fractional form of rational numbers. We continue to follow the agreement that multiplications and
divisions are done first and then the additions and subtractions, unless parentheses indicate
otherwise. The following examples illustrate a variety of situations that involve both the decimal form and fractional form of rational numbers.
Classroom Example
Simplify
5.6 Ϭ (Ϫ8) ϩ 3(4.2) Ϫ
(0.28) Ϭ (Ϫ0.7).
EXAMPLE 1
Simplify 6.3 Ϭ 7 ϩ (4) (2.1) Ϫ (0.24) Ϭ (Ϫ0.4).
Solution
6.3 Ϭ 7 ϩ (4)(2.1) Ϫ (0.24) Ϭ (Ϫ0.4) ϭ 0.9 ϩ 8.4 Ϫ (Ϫ0.6)
ϭ 0.9 ϩ 8.4 ϩ 0.6
ϭ 9.9
Classroom Example
2
1
3
Evaluate x Ϫ y for x ϭ
3
5
7
and y ϭ Ϫ2.
EXAMPLE 2
3
1
5
Evaluate a Ϫ b for a ϭ and b ϭ Ϫ1.
5
7
2
Solution
3
1
3 5
1
a Ϫ b ϭ a b Ϫ 1Ϫ12
5
7
5 2
7
3
1
ϭ ϩ
2
7
21
2
ϭ
ϩ
14
14
ϭ
Classroom Example
1
1
3
Evaluate a ϩ a Ϫ a
4
3
2
5
for a ϭϪ .
14
for a ϭ
5
and b ϭ Ϫ1
2
23
14
EXAMPLE 3
1
2
1
3
Evaluate x ϩ x Ϫ x for x ϭ Ϫ .
2
3
5
4
Solution
First, let’s combine similar terms by using the distributive property.
1
2
1
1
2
1
x ϩ x Ϫ x ϭ a ϩ Ϫ bx
2
3
5
2
3
5
15
20
6
ϭ a
ϩ
Ϫ bx
30
30
30
29
ϭ
x
30
Now we can evaluate.
29
29
3
3
xϭ
aϪ b when x ϭ Ϫ
30
30
4
4
1
29
3
29
ϭ
aϪ b ϭ Ϫ
30
4
40
10
66
Chapter 2 • Real Numbers
Classroom Example
Evaluate 4a ϩ 5b for a ϭ 2.3 and
b ϭ 1.4.
EXAMPLE 4
Evaluate 2x ϩ 3y for x ϭ 1.6 and y ϭ 2.7.
Solution
2x ϩ 3y ϭ 2(1.6) ϩ 3(2.7) when x ϭ 1.6 and y ϭ 2.7
ϭ 3.2 ϩ 8.1 ϭ 11.3
Classroom Example
Evaluate 1.5d Ϫ 0.8d ϩ 0.5d ϩ 0.2d
for d ϭ 0.4.
EXAMPLE 5
Evaluate 0.9x ϩ 0.7x Ϫ 0.4x ϩ 1.3x for x ϭ 0.2.
Solution
First, let’s combine similar terms by using the distributive property.
0.9x ϩ 0.7x Ϫ 0.4x ϩ 1.3x ϭ (0.9 ϩ 0.7 Ϫ 0.4 ϩ 1.3)x ϭ 2.5x
Now we can evaluate.
2.5x ϭ (2.5)(0.2) for x ϭ 0.2
ϭ 0.5
Classroom Example
A stain glass artist is putting together a design. She has five pieces of
glass whose lengths are 2.4 cm,
3.26 cm, 1.35 cm, 4.12 cm, and
0.7 cm. If the pieces are set side by
side, what will be their combined
length?
EXAMPLE 6
A layout artist is putting together a group of images. She has four images whose widths are
1.35 centimeters, 2.6 centimeters, 5.45 centimeters, and 3.2 centimeters. If the images are set
side by side, what will be their combined width?
Solution
To find the combined width, we need to add the widths.
1.35
2.6
5.45
ϩ3.20
12.60
The combined width would be 12.6 centimeters.
Concept Quiz 2.3
For Problems 1–10, answer true or false.
1. A rational number can be defined as any number that has a terminating or repeating
decimal representation.
2. A repeating decimal has a block of digits that repeat only once.
3. Every irrational number is also classified as a real number.
4. The rational numbers along with the irrational numbers form the set of natural
numbers.
5. 0.141414… is a rational number.
6.
7.
8.
9.
10.
Ϫ15 is real, irrational, and negative.
0.35 is real, rational, integer, and positive.
The reciprocal of c, where c 0, is also the multiplicative inverse of c.
Any number multiplied by its multiplicative inverse gives a result of 0.
Zero does not have a multiplicative inverse.
67
2.3 • Real Numbers and Algebraic Expressions
Problem Set 2.3
For Problems 1–8, classify the real numbers by tracing
down the diagram on p. 60. (Objective 1)
1. Ϫ2
51. (0.96) Ϭ (Ϫ0.8) ϩ 6(Ϫ1.4) Ϫ 5.2
52. (Ϫ2.98) Ϭ 0.4 Ϫ 5(Ϫ2.3) ϩ 1.6
2. 1/3
53. 5(2.3) Ϫ 1.2 Ϫ 7.36 Ϭ 0.8 ϩ 0.2
3. 25
54. 0.9(12) Ϭ 0.4 Ϫ 1.36 Ϭ 17 ϩ 9.2
4. Ϫ0.09090909 . . .
For Problems 55–68, simplify each algebraic expression by
combining similar terms. (Objective 3)
5. 0.16
6. Ϫ23
55. x Ϫ 0.4x Ϫ 1.8x
7. Ϫ8/7
56. Ϫ2x ϩ 1.7x Ϫ 4.6x
8. 0.125
57. 5.4n Ϫ 0.8n Ϫ 1.6n
For Problems 9 – 40, perform the indicated operations.
(Objective 2)
9. 0.37 ϩ 0.25
50. Ϫ5(0.9) Ϫ 0.6 ϩ 4.1(6) Ϫ 0.9
10. 7.2 ϩ 4.9
58. 6.2n Ϫ 7.8n Ϫ 1.3n
59. Ϫ3t ϩ 4.2t Ϫ 0.9t ϩ 0.2t
60. 7.4t Ϫ 3.9t Ϫ 0.6t ϩ 4.7t
11. 2.93 Ϫ 1.48
12. 14.36 Ϫ 5.89
13. (7.6) ϩ (Ϫ3.8)
14. (6.2) ϩ (Ϫ2.4)
15. (Ϫ4.7) ϩ 1.4
16. (Ϫ14.1) ϩ 9.5
17. Ϫ3.8 ϩ 11.3
18. Ϫ2.5 ϩ 14.8
19. 6.6 Ϫ (Ϫ1.2)
20. 18.3 Ϫ (Ϫ7.4)
21. Ϫ11.5 Ϫ (Ϫ10.6)
22. Ϫ14.6 Ϫ (Ϫ8.3)
23. Ϫ17.2 Ϫ (Ϫ9.4)
24. Ϫ21.4 Ϫ (Ϫ14.2)
25. (0.4)(2.9)
26. (0.3)(3.6)
27. (Ϫ0.8)(0.34)
28. (Ϫ0.7)(0.67)
29. (9)(Ϫ2.7)
30. (8)(Ϫ7.6)
31. (Ϫ0.7)(Ϫ64)
32. (Ϫ0.9)(Ϫ56)
33. (Ϫ0.12)(Ϫ0.13)
34. (Ϫ0.11)(Ϫ0.15)
For Problems 69–82, evaluate each algebraic expression for
the given values of the variables. Don’t forget that for some
problems it might be helpful to combine similar terms first
and then to evaluate. (Objective 4)
35. 1.56 Ϭ 1.3
36. 7.14 Ϭ 2.1
69. x ϩ 2y ϩ 3z
37. 5.92 Ϭ (Ϫ0.8)
38. Ϫ2.94 Ϭ 0.6
3
1
1
for x ϭ , y ϭ , and z ϭ Ϫ
4
3
6
39. Ϫ0.266 Ϭ (Ϫ0.7)
40. Ϫ0.126 Ϭ (Ϫ0.9)
70. 2x Ϫ y Ϫ 3z
2
3
1
for x ϭ Ϫ , y ϭ Ϫ , and z ϭ
5
4
2
61. 3.6x Ϫ 7.4y Ϫ 9.4x ϩ 10.2y
62. 5.7x ϩ 9.4y Ϫ 6.2x Ϫ 4.4y
63. 0.3(x Ϫ 4) ϩ 0.4(x ϩ 6) Ϫ 0.6x
64. 0.7(x ϩ 7) Ϫ 0.9(x Ϫ 2) ϩ 0.5x
65. 6(x Ϫ 1.1) Ϫ 5(x Ϫ 2.3) Ϫ 4(x ϩ 1.8)
66. 4(x ϩ 0.7) Ϫ 9(x ϩ 0.2) Ϫ 3(x Ϫ 0.6)
67. 5(x Ϫ 0.5) ϩ 0.3(x Ϫ 2) Ϫ 0.7(x ϩ 7)
68. Ϫ8(x Ϫ 1.2) ϩ 6(x Ϫ 4.6) ϩ 4(x ϩ 1.7)
For Problems 41– 54, simplify each of the numerical
expressions. (Objective 2)
71.
3
2
7
yϪ yϪ y
5
3
15
41. 16.5 Ϫ 18.7 ϩ 9.4
72.
1
2
3
xϩ xϪ x
2
3
4
42. 17.7 ϩ 21.2 Ϫ 14.6
43. 0.34 Ϫ 0.21 Ϫ 0.74 ϩ 0.19
44. Ϫ5.2 ϩ 6.8 Ϫ 4.7 Ϫ 3.9 ϩ 1.3
for y ϭ Ϫ
for x ϭ
5
2
4
3
7
8
73. Ϫx Ϫ 2y ϩ 4z
for x ϭ 1.7, y ϭ Ϫ2.3, and z ϭ 3.6
for x ϭ Ϫ2.9, y ϭ 7.4, and z ϭ Ϫ6.7
45. 0.76(0.2 ϩ 0.8)
46. 9.8(1.8 Ϫ 0.8)
74. Ϫ2x ϩ y Ϫ 5z
47. 0.6(4.1) ϩ 0.7(3.2)
48. 0.5(74) Ϫ 0.9(87)
75. 5x Ϫ 7y
for x ϭ Ϫ7.8 and y ϭ 8.4
76. 8x Ϫ 9y
for x ϭ Ϫ4.3 and y ϭ 5.2
49. 7(0.6) ϩ 0.9 Ϫ 3(0.4) ϩ 0.4
11
12
68
Chapter 2 • Real Numbers
77. 0.7x ϩ 0.6y
for x ϭ Ϫ2 and y ϭ 6
78. 0.8x ϩ 2.1y
for x ϭ 5 and y ϭ Ϫ9
87. The total length of the four sides of a square is 18.8 centimeters. How long is each side of the square?
82. 5x Ϫ 2 ϩ 6x ϩ 4 for x ϭ Ϫ1.1
88. When the market opened on Monday morning, Garth
bought some shares of a stock at $13.25 per share. The
daily changes in the market for that stock for the week
were 0.75, Ϫ1.50, 2.25, Ϫ0.25, and Ϫ0.50. What was
the value of one share of that stock when the market
closed on Friday afternoon?
83. Tanya bought 400 shares of one stock at $14.78 per
share, and 250 shares of another stock at $16.36 per
share. How much did she pay for the 650 shares?
89. Victoria bought two pounds of Gala apples at $1.79 per
pound and three pounds of Fuji apples at $0.99 per
pound. How much did she spend for the apples?
84. On a trip Brent bought the following amounts of gasoline: 9.7 gallons, 12.3 gallons, 14.6 gallons, 12.2 gallons, 13.8 gallons, and 15.5 gallons. How many gallons
of gasoline did he purchase on the trip?
90. In 2005 the average speed of the winner of the Daytona
500 was 135.173 miles per hour. In 1978 the average
speed of the winner was 159.73 miles per hour. How
much faster was the average speed of the winner in 1978
compared to the winner in 2005?
79. 1.2x ϩ 2.3x Ϫ 1.4x Ϫ 7.6x
80. 3.4x Ϫ 1.9x ϩ 5.2x
for x ϭ Ϫ2.5
for x ϭ 0.3
81. Ϫ3a Ϫ 1 ϩ 7a Ϫ 2 for a ϭ 0.9
85. Kathrin has a piece of copper tubing that is 76.4 centimeters long. She needs to cut it into four pieces of
equal length. Find the length of each piece.
86. On a trip Biance filled the gasoline tank and noted that
the odometer read 24,876.2 miles. After the next filling
the odometer read 25,170.5 miles. It took 13.5 gallons of
gasoline to fill the tank. How many miles per gallon did
she get on that tank of gasoline?
91. Andrea’s automobile averages 25.4 miles per gallon.
With this average rate of fuel consumption, what distance should she be able to travel on a 12.7-gallon tank
of gasoline?
92. Use a calculator to check your answers for Problems 41– 54.
Thoughts Into Words
93. At this time how would you describe the difference
between arithmetic and algebra?
95. Do you think that 222 is a rational or an irrational number? Defend your answer.
94. How have the properties of the real numbers been used
thus far in your study of arithmetic and algebra?
Further Investigations
96. Without doing the actual dividing, defend the state1
ment, “ produces a repeating decimal.” [Hint: Think
7
about the possible remainders when dividing by 7.]
97. Express each of the following in repeating decimal
form.
(a)
1
7
(b)
2
7
(c)
4
9
(d)
5
6
(e)
3
11
(f)
1
12
98. (a) How can we tell that
5
will produce a termina16
ting decimal?
(b) How can we tell that
7
will not produce a termi15
nating decimal?
(c) Determine which of the following will produce
7 11 5 7 11 13 17
a terminating decimal: , , , , , , ,
8 16 12 24 75 32 40
11 9 3
, , .
30 20 64
2.4 • Exponents
Answers to the Concept Quiz
1. True
2. False
3. True
4. False
9. False
10. True
2.4
5. True
6. True
7. False
69
8. True
Exponents
OBJECTIVES
1
Know the deﬁnition and terminology for exponential notation
2
Simplify numerical expressions that involve exponents
3
Simplify algebraic expressions by combining similar terms
4
Reduce algebraic fractions involving exponents
5
Add, subtract, multiply, and divide algebraic fractions
6
Evaluate algebraic expressions that involve exponents
We use exponents to indicate repeated multiplication. For example, we can write
5 и 5 и 5 as 53, where the 3 indicates that 5 is to be used as a factor 3 times. The following
general definition is helpful:
Deﬁnition 2.4
If n is a positive integer, and b is any real number, then
t
bn ϭ bbb . . . b
n factors of b
We refer to the b as the base and n as the exponent. The expression bn can be read as “b to
the nth power.” We frequently associate the terms squared and cubed with exponents of
2 and 3, respectively. For example, b2 is read as “b squared” and b3 as “b cubed.” An exponent of 1 is usually not written, so b1 is written as b. The following examples further clarify
the concept of an exponent.
23 ϭ 2 и 2
и2ϭ8
35 ϭ 3 и 3
и 3 и 3 и 3 ϭ 243 a 2 b
(Ϫ5) 2 ϭ (Ϫ5)(Ϫ5) ϭ 25
(0.6) 2 ϭ (0.6)(0.6) ϭ 0.36
1
4
ϭ
1
2
1
1
1
1
и 2 и 2 и 2 ϭ 16
Ϫ52 ϭ Ϫ(5 и 5) ϭ Ϫ25
We especially want to call your attention to the last two examples. Notice that (Ϫ5)2 means
that Ϫ5 is the base, which is to be used as a factor twice. However, Ϫ52 means that 5 is the
base, and after 5 is squared, we take the opposite of that result.
Exponents provide a way of writing algebraic expressions in compact form. Sometimes
we need to change from the compact form to an expanded form as these next examples demonstrate.
x4 ϭ x и x
(2x) 3 ϭ (2x)(2x)(2x)
иxиx
2y3 ϭ 2 и y и y и y
(Ϫ2x) 3 ϭ (Ϫ2x)(Ϫ2x)(Ϫ2x)
Ϫ3x5 ϭ Ϫ3 и x и x и x и x и x Ϫx2 ϭ Ϫ(x и x)
a2 ϩ b2 ϭ a и a ϩ b и b
70
Chapter 2 • Real Numbers
At other times we need to change from an expanded form to a more compact form using the
exponent notation.
и x и x ϭ 3x2
2 и 5 и x и x и x ϭ 10x3
3 и 4 и x и x и y ϭ 12x2y
7 и a и a и a и b и b ϭ 7a3b2
(2x)(3y) ϭ 2 и x и 3 и y ϭ 2 и 3 и x и y ϭ 6xy
(3a2 )(4a) ϭ 3 и a и a и 4 и a ϭ 3 и 4 и a и a и a ϭ 12a3
(Ϫ2x)(3x) ϭ Ϫ2 и x и 3 и x ϭ Ϫ2 и 3 и x и x ϭ Ϫ6x2
3
The commutative and associative properties for multiplication allowed us to rearrange and
regroup factors in the last three examples above.
The concept of exponent can be used to extend our work with combining similar terms,
operating with fractions, and evaluating algebraic expressions. Study the following examples
very carefully; they will help you pull together many ideas.
Classroom Example
Simplify 5y3 ϩ 4y3 Ϫ 3y3 by
combining similar terms.
EXAMPLE 1
Simplify 4x2 ϩ 7x2 Ϫ 2x2 by combining similar terms.
Solution
By applying the distributive property, we obtain
4x2 ϩ 7x2 Ϫ 2x2 ϭ (4 ϩ 7 Ϫ 2)x2
ϭ 9x2
Classroom Example
Simplify 6m2 Ϫ 5n3 ϩ 2m2 Ϫ 12n3
by combining similar terms.
EXAMPLE 2
Simplify -8x3 ϩ 9y2 ϩ 4x3 Ϫ 11y2 by combining similar terms.
Solution
By rearranging terms and then applying the distributive property we obtain
Ϫ8x3 ϩ 9y2 ϩ 4x3 Ϫ 11y2 ϭ Ϫ8x3 ϩ 4x3 ϩ 9y2 Ϫ 11y2
ϭ (Ϫ8 ϩ 4)x3 ϩ (9 Ϫ 11)y2
ϭ Ϫ4x3 Ϫ 2y2
Classroom Example
Simplify 6a4 Ϫ 7a Ϫ 7a4 ϩ 5a.
EXAMPLE 3
Simplify -7x2 ϩ 4x ϩ 3x2 Ϫ 9x .
Solution
Ϫ7x2 ϩ 4x ϩ 3x2 Ϫ 9x ϭ Ϫ7x2 ϩ 3x2 ϩ 4x Ϫ 9x
ϭ (Ϫ7 ϩ 3)x2 ϩ (4 Ϫ 9)x
ϭ Ϫ4x2 Ϫ 5x
As soon as you feel comfortable with this process of combining similar terms, you may
want to do some of the steps mentally. Then your work may appear as follows.
9a2 ϩ 6a2 Ϫ 12a2 ϭ 3a2
6x2 ϩ 7y2 Ϫ 3x2 Ϫ 11y2 ϭ 3x2 Ϫ 4y2
7x2y ϩ 5xy2 Ϫ 9x2y ϩ 10xy2 ϭ -2x2y ϩ 15xy2
2x3 Ϫ 5x2 Ϫ 10x Ϫ 7x3 ϩ 9x2 Ϫ 4x ϭ - 5x3 ϩ 4x2 Ϫ 14x