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3: Real Numbers and Algebraic Expressions

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.

If a and b are real numbers, then

aϩbϭbϩa

Commutative Property of Multiplication

If a and b are real numbers, then

ab ϭ ba

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

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

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

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

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

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

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

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

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

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3: Real Numbers and Algebraic Expressions

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