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6: Integral Exponents and Scientific Notation

# 6: Integral Exponents and Scientific Notation

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5.6 • Integral Exponents and Scientiﬁc Notation

213

Property 5.5

If m and n are positive integers, and a and b are real numbers, except b ϶ 0 whenever it

appears in a denominator, then

1. bn

и bm ϭ bnϩm

2. (bn)m ϭ bmn

3. (ab)n ϭ anbn

a n an

4. a b ϭ n

b

b

5.

Part 4 has not been stated previously

bn

ϭ bnϪm

bm

When n Ͼ m

bn

ϭ1

bm

When n ϭ m

Property 5.5 pertains to the use of positive integers as exponents. Zero and the negative integers can also be used as exponents. First, let’s consider the use of 0 as an exponent. We want

to use 0 as an exponent in such a way that the basic properties of exponents will continue to

hold. Consider the example x4 # x0. If part 1 of Property 5.5 is to hold, then

x4

# x0 ϭ x4ϩ0 ϭ x4

Note that x0 acts like 1 because x4

# x0 ϭ x4. This suggests the following definition.

Deﬁnition 5.2

If b is a nonzero real number, then

b0 ϭ 1

According to Definition 5.2 the following statements are all true.

40 ϭ 1

1Ϫ6282 0 ϭ 1

4 0

a b ϭ1

7

n0 ϭ 1,  n

0

(x y ) ϭ 1,  x ϶ 0 and y ϶ 0

2 5 0

A similar line of reasoning indicates how negative integers should be used as exponents.

Consider the example x3 и xϪ3. If part 1 of Property 5.5 is to hold, then

x3

и xϪ3 ϭ x3ϩ(Ϫ3) ϭ x0 ϭ 1

Thus xϪ3 must be the reciprocal of x3 because their product is 1; that is,

xϪ3 ϭ

1

x3

This process suggests the following definition.

214

Chapter 5 • Exponents and Polynomials

Deﬁnition 5.3

If n is a positive integer, and b is a nonzero real number, then

1

bϪn ϭ n

b

According to Definition 5.3, the following statements are all true.

x Ϫ6 ϭ

1

x6

2Ϫ3 ϭ

1

1

ϭ

3

8

2

10Ϫ2 ϭ

1

1

ϭ

or  0.01

100

102

1

1

ϭ

ϭ x4

Ϫ4

1

x

x4

2 Ϫ2

1

1

9

a b ϭ

ϭ ϭ

3

4

4

2 2

a b

9

3

2 Ϫ2

3 2

ϭ a b . In other words, to raise a fraction

3

2

to a negative power, we can invert the fraction and raise it to the corresponding positive power.

Remark: Note in the last example that a b

We can verify (we will not do so in this text) that all parts of Property 5.5 hold for all

integers. In fact, we can replace part 5 with this statement.

Replacement for part 5 of Property 5.5

bn

ϭ bnϪm

bm

for all integers n and m

The next examples illustrate the use of this new concept. In each example, we simplify the

original expression and use only positive exponents in the final result.

x2

1

ϭ x2Ϫ5 ϭ xϪ3 ϭ 3

5

x

x

aϪ3

ϭ aϪ3Ϫ 1Ϫ72 ϭ aϪ3ϩ7 ϭ a4

aϪ7

yϪ5

Ϫ2

y

ϭ yϪ5Ϫ 1Ϫ22 ϭ yϪ5ϩ2 ϭ yϪ3 ϭ

1

y3

xϪ6

ϭ xϪ6Ϫ 1Ϫ62 ϭ xϪ6ϩ6 ϭ x0 ϭ 1

xϪ6

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5.6 • Integral Exponents and Scientiﬁc Notation

215

The properties of exponents provide a basis for simplifying certain types of numerical

expressions, as the following examples illustrate.

2Ϫ4

# 26 ϭ 2Ϫ4ϩ6 ϭ 22 ϭ 4

105

и 10Ϫ6 ϭ 105ϩ (Ϫ6) ϭ 10Ϫ1 ϭ 10     or    0.1

1

102

ϭ 102Ϫ(Ϫ2) ϭ 102ϩ2 ϭ 104 ϭ 10,000

10Ϫ2

(2Ϫ3)Ϫ2 ϭ 2Ϫ3(Ϫ2) ϭ 26 ϭ 64

Having the use of all integers as exponents also expands the type of work that we can do

with algebraic expressions. In each of the following examples we simplify a given expression

and use only positive exponents in the final result.

x8xϪ2 ϭ x8ϩ(Ϫ2) ϭ x6

aϪ4aϪ3 ϭ aϪ4ϩ(Ϫ3) ϭ aϪ7 ϭ

(yϪ3)4 ϭ yϪ3(4) ϭ yϪ12 ϭ

1

a7

1

y12

(xϪ2y4)Ϫ3 ϭ (xϪ2)Ϫ3(y4)Ϫ3 ϭ x6yϪ12 ϭ

a

x6

y12

xϪ1 Ϫ2 (xϪ1)Ϫ2

x2

b

ϭ

ϭ

ϭ x2y4

y2

(y2)Ϫ2

yϪ4

(4xϪ2)(3xϪ1) ϭ 12xϪ2ϩ(Ϫ1) ϭ 12xϪ3 ϭ

a

12

x3

12xϪ6 Ϫ2

b ϭ (2xϪ6Ϫ(Ϫ2))Ϫ2 ϭ (2xϪ4)Ϫ2

6xϪ2

Divide the coefficients

12

ϭ2

6

ϭ (2)Ϫ2(xϪ4)Ϫ2

ϭa

1

x8

b(x8) ϭ

2

4

2

Scientiﬁc Notation

Many scientific applications of mathematics involve the use of very large and very small

numbers. For example:

The speed of light is approximately 29,979,200,000 centimeters per second.

A light year (the distance light travels in 1 year) is approximately 5,865,696,000,000 miles.

A gigahertz equals 1,000,000,000 hertz.

The length of a typical virus cell equals 0.000000075 of a meter.

The length of a diameter of a water molecule is 0.0000000003 of a meter.

Working with numbers of this type in standard form is quite cumbersome. It is much more

convenient to represent very small and very large numbers in scientific notation, sometimes

called scientific form. A number is in scientific notation when it is written as the product of

a number between 1 and 10 (including 1) and an integral power of 10. Symbolically, a number in scientific notation has the form 1N2110k 2 , where 1 Յ N Ͻ 10, and k is an integer. For

example, 621 can be written as 16.2121102 2 , and 0.0023 can be written as 12.32110Ϫ3 2 .

To switch from ordinary notation to scientific notation, you can use the following

procedure.

216

Chapter 5 • Exponents and Polynomials

Write the given number as the product of a number greater than or equal to 1 and less

than 10, and an integral power of 10. To determine the exponent of 10, count the

number of places that the decimal point moved when going from the original number

to the number greater than or equal to 1 and less than 10. This exponent is (a) negative

if the original number is less than 1, (b) positive if the original number is greater than 10,

and (c) zero if the original number itself is between 1 and 10.

Thus we can write

0.000179 ϭ (1.79)(10Ϫ4)

8175 ϭ (8.175)(103)

3.14 ϭ (3.14)(100)

According to part (a) of the procedure

According to part (b)

According to part (c)

We can express the applications given earlier in scientific notation as follows:

Speed of light: 29,979,200,000 ϭ (2.99792)(1010 ) centimeters per second

Light year: 5,865,696,000,000 ϭ (5.865696)(1012) miles

Gigahertz: 1,000,000,000 ϭ (1)(10 9) hertz

Length of a virus cell: 0.000000075 ϭ (7.5)(10Ϫ8) meter

Length of the diameter of a water molecule ϭ 0.0000000003

ϭ (3)(10Ϫ10) meter

To switch from scientific notation to ordinary decimal notation you can use the following procedure.

Move the decimal point the number of places indicated by the exponent of 10. The decimal

point is moved to the right if the exponent is positive and to the left if it is negative.

Thus we can write

Two zeros are needed for place value purposes

(4.71)(104) ϭ 47,100

Ϫ2

(1.78)(10 ) ϭ 0.0178

One zero is needed for place value purposes

The use of scientific notation along with the properties of exponents can make some

arithmetic problems much easier to evaluate. The next examples illustrate this point.

Classroom Example

Evaluate (6000)(0.00072).

EXAMPLE 1

Evaluate (4000)(0.000012).

Solution

(4000)(0.000012) ϭ

ϭ

ϭ

ϭ

Classroom Example

Evaluate

EXAMPLE 2

(4)(103)(1.2)(10Ϫ5)

(4)(1.2)(103)(10Ϫ5)

(4.8)(10Ϫ2)

0.048

Evaluate

840,000

.

0.024

960,000

.

0.032

Solution

960,000

(9.6)(105)

ϭ

0.032

(3.2)(10Ϫ2)

ϭ (3)(107)

ϭ 30,000,000

105

ϭ 105Ϫ(Ϫ2) ϭ 107

10Ϫ2

5.6 • Integral Exponents and Scientiﬁc Notation

EXAMPLE 3

Classroom Example

Evaluate

Evaluate

(7000)(0.0000009)

.

(0.0012)(30,000)

217

(6000)(0.00008)

.

(40,000)(0.006)

Solution

(6000)(0.00008)

(6)(103)(8)(10Ϫ5)

ϭ

(40,000)(0.006)

(4)(104)(6)(10Ϫ3)

ϭ

(48)(10Ϫ2)

(24)(101)

ϭ (2)(10Ϫ3)

ϭ 0.002

10Ϫ2

ϭ 10Ϫ2Ϫ1 ϭ 10Ϫ3

101

Concept Quiz 5.6

For Problems 1–10, answer true or false.

1. Any nonzero number raised to the zero power is equal to one.

2. The algebraic expression xϪ2 is the reciprocal of x2 for x ϶ 0.

3. To raise a fraction to a negative exponent, we can invert the fraction and raise it to the

corresponding positive exponent.

1

4. Ϫ3 ϭ yϪ3

y

5. A number in scientific notation has the form (N)(10k) where 1 Յ N Ͻ 10, and k is any

real number.

6. A number is less than zero if the exponent is negative when the number is written in

scientific notation.

1

7. Ϫ2 ϭ x2

x

8.

10Ϫ2

ϭ 100

10Ϫ4

9. (3.11)(10Ϫ2) ϭ 311

10. (5.24)(10Ϫ1) ϭ 0.524

Problem Set 5.6

For Problems 1– 30, evaluate each numerical expression.

(Objective 1)

1. 3Ϫ2

2. 2Ϫ5

3. 4Ϫ3

3 Ϫ1

5. a b

2

3 Ϫ2

6. a b

4

1

7. Ϫ4

2

1

8. Ϫ1

3

4. 5Ϫ2

15. Ϫ(3Ϫ2)

17.

1

3 Ϫ3

a b

4

и 2Ϫ9

21. 36 # 3Ϫ3

19. 26

16. Ϫ(2Ϫ2)

18.

1

3 Ϫ4

a b

2

и 3Ϫ2

22. 2Ϫ7 и 22

20. 35

4 0

9. aϪ b

3

1 Ϫ3

10. aϪ b

2

23.

102

10Ϫ1

24.

101

10Ϫ3

2 Ϫ3

11. aϪ b

3

12. (Ϫ16)0

25.

10Ϫ1

102

26.

10Ϫ2

10Ϫ2

13. (Ϫ2)Ϫ2

14. (Ϫ3)Ϫ2

27. (2Ϫ1 и 3Ϫ2)Ϫ1

28. (3Ϫ1 и 4Ϫ2)Ϫ1

218

Chapter 5 • Exponents and Polynomials

29. a

4Ϫ1 Ϫ2

b

3

30. a

3 Ϫ3

b

2Ϫ1

For Problems 31– 84, simplify each algebraic expression and

express your answers using positive exponents only.

(Objective 1)

31. x6xϪ1

32. xϪ2x7

33. nϪ4n2

34. nϪ8n3

Ϫ2 Ϫ3

Ϫ4 Ϫ6

35. a a

36. a a

37. (2x )(4x )

38. (5xϪ4)(6x7)

39. (3xϪ6)(9x2)

40. (8xϪ8)(4x2)

41. (5yϪ1)(Ϫ3yϪ2)

42. (Ϫ7yϪ3)(9yϪ4)

43. (8xϪ4)(12x4)

44. (Ϫ3xϪ2)(Ϫ6x2)

3

45.

Ϫ2

x7

xϪ3

46.

Ϫ1

x2

xϪ4

Ϫ2

47.

n

n3

48.

n

n5

49.

4nϪ1

2nϪ3

50.

12nϪ2

3nϪ5

51.

Ϫ24xϪ6

8xϪ2

52.

56xϪ5

Ϫ7xϪ1

53.

Ϫ52yϪ2

Ϫ13yϪ2

54.

Ϫ91yϪ3

Ϫ7yϪ3

55. (xϪ3)Ϫ2

56. (xϪ1)Ϫ5

57. (x2)Ϫ2

58. (x3)Ϫ1

59. (x3y4)Ϫ1

60. (x4yϪ2)Ϫ2

61. (xϪ2yϪ1)3

62. (xϪ3yϪ4)2

63. (2nϪ2)3

64. (3nϪ1)4

65. (4n3)Ϫ2

66. (2n2)Ϫ3

67. (3aϪ2)4

68. (5aϪ1)2

69. (5xϪ1)Ϫ2

70. (4xϪ2)Ϫ2

71. 12xϪ2yϪ1 2 Ϫ1

72. (3x2yϪ3)Ϫ2

x2 Ϫ1

73. a b

y

y2 Ϫ2

74. a 3 b

x

aϪ1 Ϫ4

75. a 2 b

b

a3 Ϫ3

76. a Ϫ2 b

b

xϪ1 Ϫ2

77. a Ϫ3 b

y

xϪ3 Ϫ1

78. a Ϫ4 b

y

x2 Ϫ1

79. a 3 b

x

x4 Ϫ2

80. a b

x

81. a

2xϪ1 Ϫ3

b

xϪ2

82. a

3xϪ2 Ϫ1

b

xϪ5

83. a

18xϪ1 Ϫ2

b

9x

84. a

35x2 Ϫ1

b

7xϪ1

85. The U.S. Social Security Administration pays approximately \$492,000,000,000 in monthly benefits to all

beneficiaries. Write this number in scientific notation.

86. In 2008 approximately 5,419,200,000 pennies were

made. Write, in scientific notation, the number of pennies made.

87. The thickness of a dollar bill is 0.0043 inches. Write this

number in scientific notation.

88. The average length of an Ebola virus cell is approximately 0.0000002 meters. Write this number in scientific

notation.

89. The diameter of Jupiter is approximately 89,000 miles.

Write this number in scientific notation.

90. Avogadro’s Number, which has a value of approximately

602,200,000,000,000,000,000,000, refers to the calculated value of the number of molecules in a gram mole of

any chemical substance. Write Avogadro’s Number in

scientific notation.

91. The cooling fan for an old computer hard drive has a

thickness of 0.025 meters. Write this number in scientific

notation.

92. A sheet of 20-weight bond paper has a thickness of

0.097 millimeters. Write the thickness in scientific

notation.

93. According to the 2004 annual report for Coca-Cola, the

Web site CokePLAY.com was launched in Korea, and it

was visited by 11,000,000 people in its first six months.

Write the number of hits for the Web site in scientific

notation.

94. In 2005, Wal-Mart reported that they served 138,000,000

customers worldwide each week. Write, in scientific notation, the number of customers for each week.

For Problems 95 – 106, write each number in standard decimal form; for example, (1.4)(103) ϭ 1400. (Objective 3)

95. (8)(103)

96. (6)(102)

97. (5.21)(104)

98. (7.2)(103)

99. (1.14)(107)

100. (5.64)(108)

101. (7)(10Ϫ2)

102. (8.14)(10Ϫ1)

103. (9.87)(10Ϫ4)

104. (4.37)(10Ϫ5)

105. (8.64)(10Ϫ6)

106. (3.14)(10Ϫ7)

5.6 • Integral Exponents and Scientiﬁc Notation

For Problems 107–118, use scientific notation and the properties of exponents to evaluate each numerical expression.

(Objective 4)

107. (0.007)(120)

109. (5,000,000)(0.00009)

110. (800,000)(0.0000006)

6000

0.0015

112.

480

0.012

113.

0.00086

4300

114.

0.0057

30,000

115.

0.00039

0.0013

116.

0.0000082

0.00041

117.

10.0008210.072

120,000210.00042

118.

For Problems 119 –122, convert the numbers to scientific

notation and compute the answer. (Objectives 2 and 4)

119. The U.S. Social Security Administration pays approximately \$42,000,000,000 in benefits to all beneficiaries. If

there are 48,000,000 beneficiaries, find the average dollar

amount each beneficiary receives.

108. (0.0004)(13)

111.

219

120. In 1998 approximately 10,200,000,000 pennies were

made. If the population was 270,000,000, find the average number of pennies produced per person. Round the

answer to the nearest whole number.

121. The thickness of a dollar bill is 0.0043 inches. How tall

will a stack of 1,000,000, dollars be? Express the answer to the nearest foot.

122. The diameter of Jupiter is approximately 11 times larger

than the diameter of Earth. Find the diameter of Earth

given that Jupiter has a diameter of approximately

89,000 miles. Express the answer to the nearest mile.

10.006216002

10.0000421302

Thoughts Into Words

124. Explain the importance of scientific notation.

123. Is the following simplification process correct?

12Ϫ2 2 Ϫ1 ϭ a

1 Ϫ1

1 Ϫ1

b ϭ a b ϭ

2

4

2

1

ϭ4

1 1

a b

4

Can you suggest a better way to do the problem?

Further Investigations

125. Use your calculator to do Problems 1– 16. Be sure that

your answers are equivalent to the answers you obtained without the calculator.

2

126. Use your calculator to evaluate (140,000) . Your

answer should be displayed in scientific notation;

the format of the display depends on the particular

calculator. For example, it may look like 1.96 10 or

1.96E + 10 . Thus in ordinary notation the answer is

19,600,000,000. Use your calculator to evaluate

Answers to the Concept Quiz

1. True

2. True

3. True

4. False

9. False

10. True

5. False

each expression. Express final answers in ordinary

notation.

(a) (9000)3

(b) (4000)3

(c) (150,000)2

(d) (170,000)2

(e) (0.012)5

(f) (0.0015)4

(g) (0.006)3

(h) (0.02)6

127. Use your calculator to check your answers to Problems

107 –118.

6. False

7. True

8. True

Chapter 5 Summary

OBJECTIVE

SUMMARY

EXAMPLE

Classify polynomials by size

and degree.

Terms with variables that contain only whole

number exponents are called monomials. A

polynomial is a monomial or a finite sum of

monomials. The degree of a monomial is the

sum of the exponents of the literal factors.

The degree of a polynomial is the degree of

the term with the highest degree in the polynomial. A one-term polynomial is called a

monomial, a two-term polynomial is called

a binomial, and a three-term polynomial is

called a trinomial.

The polynomial 6x2y3 is a monomial of

degree 5.

The polynomial 3x2y2 Ϫ 7xy2 is a binomial

of degree 4.

The polynomial 8x2 ϩ 3x Ϫ1 is a trinomial

of degree 2.

The polynomial x4 ϩ 6x3 Ϫ 5x2 ϩ 12x Ϫ 2

is a polynomial of degree 4.

Addition and subtraction of polynomials is

based on using the distributive property

and combining similar terms. A polynomial

is subtracted by adding the opposite. The

opposite of a polynomial is formed by

taking the opposite of each term.

(a) Add 3x2 Ϫ 2x ϩ 1 and 4x 2 ϩ x Ϫ 5.

(b) Perform the subtraction (4x Ϫ 3) Ϫ

(2x ϩ 7).

(Section 5.1/Objective 1)

(Section 5.1/Objective 2)

Add and subtract polynomials.

(Section 5.1/Objective 3)

(Section 5.1/Objective 4)

Perform operations on

polynomials involving both

(Section 5.1/Objective 3)

(Section 5.1/Objective 4)

Apply properties of

exponents to multiply

monomials.

We can use the distributive property along

with the properties a ϭ 1(a) and Ϫa ϭ

Ϫ1(a) when adding and subtracting

polynomials.

When multiplying powers with the same

base, add the exponents. When raising a

power to a power, multiply the exponents.

(Section 5.2/Objective 1)

Apply properties of

exponents to raise a

monomial to a power.

(Section 5.2/Objective 1)

220

Solution

(a) (3x2 Ϫ 2x ϩ 1) ϩ (4x 2 ϩ x Ϫ 5)

ϭ (3x2 ϩ 4x2 ) ϩ (Ϫ2x ϩ x) ϩ (1 Ϫ 5)

ϭ (3 ϩ 4)x2 ϩ (Ϫ2 ϩ 1)x ϩ (1 Ϫ 5)

ϭ 7x2 Ϫ x Ϫ 4

(b) (4x Ϫ 3) Ϫ (2x ϩ 7)

ϭ (4x Ϫ 3) ϩ (Ϫ2x Ϫ 7)

ϭ (4x Ϫ 2x) ϩ (Ϫ 3 Ϫ 7)

ϭ 2x Ϫ 10

Perform the indicated operation:

3y ϩ 4 Ϫ (6y Ϫ 5)

Solution

3y ϩ 4 Ϫ (6y Ϫ 5)

ϭ 3y ϩ 4 Ϫ 1(6y Ϫ 5)

ϭ 3y ϩ 4 Ϫ 1(6y) Ϫ 1(Ϫ5)

ϭ 3y ϩ 4 Ϫ 6y ϩ 5 ϭ Ϫ3y ϩ 9

Find the products:

(a) (3x2)(4x3) (b) (n2)4

Solution

(a) (3x2)(4x3)

ϭ 3 и 4 и x2ϩ3

ϭ 12x5

(b) (n2)4

ϭ n2 и 4

ϭ n8

When raising a monomial to a power, raise

each factor to that power.

Simplify (6ab2)3.

Solution

(6ab2)3

ϭ 63 и a3 и b6

ϭ 216a3b6

Chapter 5 • Summary

OBJECTIVE

SUMMARY

EXAMPLE

Find the product of two

polynomials.

The distributive property along with the

properties of exponents form a basis for

multiplying polynomials.

Multiply (2x ϩ 3)(x2 Ϫ 4x ϩ 5).

The product of two binomials is perhaps the

most frequently used type of multiplication

problem. A three-step shortcut pattern,

called FOIL, is often used to find the

product of two binomials. FOIL stands for

first, outside, inside, and last.

Multiply (2x ϩ 5)(x Ϫ 3).

(Section 5.3/Objective 1)

Use the shortcut pattern

to find the product of two

binomials.

(Section 5.3/Objective 2)

Raise a binomial to a power.

(Section 5.3/Objective 1)

Use a special product pattern

to find products.

(Section 5.3/Objective 3)

Apply polynomials to

geometric problems.

(Section 5.2/Objective 3)

When raising a binomial to a power, rewrite

the problem as a product of binomials.

Then you can apply the FOIL shortcut to

multiply two binomials. Then apply the

distributive property to find the product.

When multiplying binomials you should be

able to recognize the following special

patterns and use them to find the product.

(a ϩ b)2 ϭ a2 ϩ 2ab ϩ b2

(a Ϫ b)2 ϭ a2 Ϫ 2ab ϩ b2

(a ϩ b)(a Ϫ b) ϭ a2 Ϫ b2

Polynomials can be applied to represent

perimeters, areas, and volumes of

geometric figures.

221

Solution

(2x ϩ 3)(x2 Ϫ 4x ϩ 5)

ϭ 2x (x2 Ϫ 4x ϩ 5) ϩ 3(x2 Ϫ 4x ϩ 5)

ϭ 2x3 Ϫ 8x2 ϩ 10x ϩ 3x2 Ϫ 12x ϩ 15

ϭ 2x3 Ϫ 5x2 Ϫ 2x ϩ 15

Solution

(2x ϩ 5)(x Ϫ 3)

ϭ 2x(x) ϩ (2x)(Ϫ3) ϩ (5)(x) ϩ (5)(Ϫ3)

ϭ 2x2 Ϫ 6x ϩ 5x Ϫ 15

ϭ 2x2 Ϫ x Ϫ 15

Find the product (x ϩ 4)3.

Solution

(x ϩ 4)3ϭ (x ϩ 4)(x ϩ 4)(x ϩ 4)

ϭ (x ϩ 4)(x2 ϩ 8x ϩ 16)

ϭ x(x2ϩ8xϩ16) ϩ 4(x2ϩ8xϩ16)

ϭ x3 ϩ 8x2 ϩ 16x ϩ 4x2 ϩ 32x ϩ 64

ϭ x3 ϩ 12x2 ϩ 48x ϩ 64

Find the product (5x ϩ 6)2.

Solution

(5x ϩ 6)2 ϭ (5x)2 ϩ 2(5x)(6) ϩ (6)2

ϭ 25x2 ϩ 60x ϩ 36

A piece of aluminum that is 15 inches by

20 inches has a square piece x inches on a

side cut out from two corners. Find the

area of the aluminum piece after the

corners are removed.

Solution

The area before the corners are removed is

found by applying the formula A ϭ lw.

Therefore A ϭ 15(20) ϭ 300. Each of the

two corners removed has an area of x2.

Therefore 2x2 must be subtracted from the

area of the aluminum piece. The area of

the aluminum piece after the corners are

removed is 300 Ϫ 2x2.

Apply properties of

exponents to divide

monomials.

(Section 5.4/Objective 1)

The following properties of exponents, along

with our knowledge of dividing integers,

serves as a basis for dividing monomials.

bn

ϭ bnϪm when n > m

bm

bn

ϭ 1 when n ϭ m

bm

Divide

Ϫ32 a5 b4

.

8a3 b

Solution

Ϫ32 a5 b4

ϭ Ϫ4a2 b3

8a3 b

(continued)

222

Chapter 5 • Exponents and Polynomials

OBJECTIVE

SUMMARY

Divide polynomials by

monomials.

Dividing a polynomial by a monomial is

based on the property

aϩb

a

b

ϭ ϩ

c

c

c

(Section 5.4/Objective 2)

Divide polynomials by

binomials.

(Section 5.5/Objective 1)

Apply the properties of

exponents, including negative

and zero exponents, to

simplify expressions.

(Section 5.6/Objective 1)

Write numbers expressed in

scientific notation in standard

decimal notation.

(Section 5.6/Objective 3)

Write numbers in scientific

notation.

(Section 5.6/Objective 2)

Use scientific notation to

evaluate numerical

expressions.

(Section 5.6/Objective 4)

Use the conventional long division format

from arithmetic. Arrange both the dividend

and the divisor in descending powers of the

variable. You may have to insert terms with

zero coefficients if terms with some powers

of the variable are missing.

By definition, if b is a nonzero real

number, then b0 ϭ 1.

By definition, if n is a positive integer, and

1

b is a nonzero real number, then b Ϫn ϭ n .

b

When simplifying expressions use only

positive exponents in the final result.

EXAMPLE

Divide

8x5 Ϫ 16x4 ϩ 4x3

.

4x

Solution

8x5 Ϫ16x4 ϩ 4x3 8x5

Ϫ 16x4 4x3

ϭ

ϩ

ϩ

4x

4x

4x

4x

4

3

2

ϭ 2x Ϫ 4x ϩ x

Divide (y3 Ϫ 5y Ϫ 2) Ϭ (y ϩ 2).

Solution

y2 Ϫ2y Ϫ1

y ϩ2 ͤy 3 ϩ 0y 2 Ϫ 5y Ϫ 2

y3 ϩ 2y 2

Ϫ2y 2 Ϫ5y

Ϫ2y 2 Ϫ 4y

Ϫy Ϫ2

Ϫy Ϫ2

28n Ϫ6

. Express the answer using

4n Ϫ2

positive exponents only.

Simplify

Solution

28n Ϫ6

7

ϭ 7n Ϫ6Ϫ1 Ϫ22 ϭ7n Ϫ6ϩ2 ϭ7n Ϫ4 ϭ 4

Ϫ2

4n

n

To change from scientific notation to

ordinary decimal notation, move the decimal

point the number of places indicated by the

exponent of 10. The decimal point is

moved to the right if the exponent is

positive and to the left if it is negative.

Write (3.28)(10Ϫ4) in standard decimal

notation.

To represent a number in scientific

notation, express it as a product of a

number between 1 and 10 (including 1)

and an integral power of ten.

Write 657,000,000 in scientific notation.

Scientific notation can be used to make

arithmetic problems easier to evaluate.

Solution

(3.28)(10Ϫ4) ϭ 0.000328

Solution

Count the number of decimal places from

the existing decimal point until you have

a number between 1 and 10. For the

scientific notation raise 10 to that number.

657,000,000 ϭ (6.57)(108)

Change each number to scientific notation

and evaluate the expression. Express the

answer in standard notation.

(61,000)(0.000005).

Solution

(61,000)(0.000005) ϭ (6.1)(104)(5)(10Ϫ6)

ϭ (6.1)(5)(104)(10Ϫ6)

ϭ (30.5)(10Ϫ2)

ϭ 0.305

Chapter 5 • Review Problem Set

223

Chapter 5 Review Problem Set

For Problems 1– 4, perform the additions and subtractions.

35. (x Ϫ 2)(x2 Ϫ x ϩ 6)

1. (5x2 Ϫ 6x ϩ 4) ϩ (3x2 Ϫ 7x Ϫ 2)

36. (2x Ϫ 1)(x2 ϩ 4x ϩ 7)

2. (7y2 ϩ 9y Ϫ 3) Ϫ (4y2 Ϫ 2y ϩ 6)

37. (a ϩ 5)3

3. (2x2 ϩ 3x Ϫ 4) ϩ(4x2 Ϫ 3x Ϫ 6) Ϫ (3x2 Ϫ 2x Ϫ 1)

38. (a Ϫ 6)3

4. (Ϫ3x2 Ϫ 2x ϩ 4) Ϫ(x2 Ϫ 5x Ϫ 6) Ϫ (4x2 ϩ 3x Ϫ 8)

39. (x2 Ϫ x Ϫ 1)(x2 ϩ 2x ϩ 5)

For Problems 5 –12, remove parentheses and combine similar terms.

5. 5(2x Ϫ 1) ϩ 7(x ϩ 3) Ϫ 2(3x ϩ 4)

For Problems 41– 48, perform the divisions.

41.

6. 3(2x Ϫ 4x Ϫ 5) Ϫ 5(3x Ϫ 4x ϩ 1)

2

40. (n2 ϩ 2n ϩ 4)(n2 Ϫ7n Ϫ1)

2

7. 6(y Ϫ 7y Ϫ 3) Ϫ 4(y ϩ 3y Ϫ 9)

2

2

43.

8. 3(a Ϫ 1) Ϫ 2(3a Ϫ 4) Ϫ 5(2a ϩ 7)

9. Ϫ(a ϩ 4) ϩ 5(Ϫa Ϫ 2) Ϫ 7(3a Ϫ 1)

10. Ϫ2(3n Ϫ 1) Ϫ 4(2n ϩ 6) ϩ 5(3n ϩ 4)

11. 3(n2 Ϫ 2n Ϫ 4) Ϫ 4(2n2 Ϫ n Ϫ 3)

36x4y5

42.

Ϫ3xy2

Ϫ18x4y3 Ϫ 54x6y2

6x2y2

44.

Ϫ30a5b10 ϩ 39a4b8

Ϫ3ab

45.

56x4 Ϫ 40x3 Ϫ 32x2

4x2

12. Ϫ5(Ϫn2 ϩ n Ϫ 1) ϩ 3(4n2 Ϫ 3n Ϫ 7)

Ϫ56a5b7

Ϫ8a2b3

46. (x2 ϩ 9x Ϫ 1) Ϭ (x ϩ 5)

For Problems 13–20, find the indicated products.

13. (5x2)(7x4)

2

14. (Ϫ6x3)(9x5)

2 3

15. (Ϫ4xy )(Ϫ6x y )

2 3 3

3 4

5

18. (Ϫ3xy2)2

19. 5x(7x ϩ 3)

20. (Ϫ3x2)(8x Ϫ 1)

For Problems 21– 40, find the indicated products. Be sure to

22. (3x ϩ 7)(x ϩ 1)

23. (x Ϫ 5)(x ϩ 2)

48. (2x3 Ϫ 3x2 ϩ 2x Ϫ 4) Ϭ (x Ϫ 2)

16. (2a b )(Ϫ3ab )

17. (2a b )

21. (x ϩ 9)(x ϩ 8)

47. (21x2 Ϫ 4x Ϫ 12) Ϭ (3x ϩ 2)

For Problems 49– 60, evaluate each expression.

49. 32 ϩ 22

50. (3 ϩ 2)2

51. 2Ϫ4

52. (Ϫ5)0

53. Ϫ50

54.

3 Ϫ2

55. a b

4

56.

24. (y Ϫ 4)(y Ϫ 9)

25. (2x Ϫ 1)(7x ϩ 3)

26. (4a Ϫ 7)(5a ϩ 8)

27. (3a Ϫ 5)

2

57.

1

(Ϫ2)Ϫ3

59. 30 ϩ 2Ϫ2

1

3Ϫ2

1

1 Ϫ1

a b

4

58. 2Ϫ1 ϩ 3Ϫ2

60. (2 ϩ 3)Ϫ2

28. (x ϩ 6)(2x2 ϩ 5x Ϫ 4)

For Problems 61–72, simplify each of the following, and

express your answers using positive exponents only.

29. (5n Ϫ 1)(6n ϩ 5)

61. x5xϪ8

30. (3n ϩ 4)(4n Ϫ 1)

31. (2n ϩ 1)(2n Ϫ 1)

32. (4n Ϫ 5)(4n ϩ 5)

33. (2a ϩ 7)2

34. (3a ϩ 5)2

62. (3x5)(4xϪ2)

63.

xϪ4

xϪ6

64.

xϪ6

xϪ4

65.

24a5

3aϪ1

66.

48nϪ2

12nϪ1

67. (xϪ2y)Ϫ1

68. (a2bϪ3)Ϫ2

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