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
addition and subtraction.
(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
simplify your answers.
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