Download Chapter Summary and Summary Exercises (PDF Files)

Summary
8
DEFINITION /PROCEDURE
EXAMPLE
REFERENCE
Roots and Radicals
Section 8.1
Square Roots Every positive number has two square roots.
The positive or principal square root of a number a is denoted
1a
125 5
5 is the principal square root of
25 because 52 25.
149 7
The negative square root is written as
1a
because (7)2 49.
Higher Roots Cube roots, fourth roots, and so on are denoted
by using an index and a radical. The principal nth root of a is
written as
3
127 3
3
164 4
4
181 3
Index
242 4
n
1a
2(5)2 5
3
Radical sign
Radicand
2(3)3 3
2m2 m
Radicals Containing Variables In general,
x if n is even
n
2 xn x
if n is odd
3
227x3 3x
p. 581
Simplification of Radical Expressions
Section 8.2
Simplifying radical expressions entails applying two properties
for radicals.
Product Property
n
n
n
1 ab 1 a 1 b
135 15 7
15 17
p. 591
Quotient Property
n
1a
a
n ,b 0
1b
Ab
n
p. 591
2
12
A5
15
Simplified Form for Radicals A radical is in simplified form
if the following three conditions are satisfied.
1. The radicand has no factor raised to a power greater than or
equal to the index.
218x3 29x2 2x
29x2 12x
2. No fraction appears in the radical.
© 2001 McGraw-Hill Companies
3. No radical appears in a denominator.
Note: Satisfying the third condition may require rationalizing
the denominator.
3x12x
5
15
15
A9
19
3
3
13
13 17x
A 7x
17x
17x 17x
121x
249x
2
121x
7x
p. 592
Continued
657
658
CHAPTER 8
RADICAL EXPRESSIONS
DEFINITION /PROCEDURE
EXAMPLE
REFERENCE
Operations on Radical Expressions
Radical expressions may be combined only if they are similar,
that is, if they have the same radicand with the same index.
Similar radicals are combined by application of the
distributive property.
Multiplication To multiply two radical expressions, we use
n
n
n
1 a 1 b 1 ab
and simplify the product.
If binomial expressions are involved, we use the distributive
property or the FOIL method.
Division To divide two radical expressions, rationalize the
denominator by multiplying the numerator and denominator by
the appropriate radical.
If the divisor (the denominator) is a binomial, multiply the
numerator and denominator by the conjugate of the
denominator.
Section 8.3
815 315 (8 3)15
1115
2118 412
219 2 4 12
219 12 4 12
2 3 12 4 12
612 412 (6 4)12
212
p. 601
13x 26x 218x
29x2 2x
29x2 12x
2
3
3x12x
12(5 18) 12 5
12 18
5 12 4
(3 12)(5 12)
15 312 512 2
13 212
5 12
512
5
18
18 12
116
512
4
p. 604
Note: 3 15 is the conjugate
of 3 15.
2
2(3 15)
3 15 (3 15)(3 15)
2(3 15)
4
3 15
2
Solving Radical Equations
Power Property of Equality
If a b then an bn
p. 607
Section 8.4
If 1x 1 5
then (1x 1)2 52
x 1 25
If an equation involves two radicals, rewrite the equation so
that there is one radical on each side and then use the power
property to solve it.
p. 615
Given 1x 1x 7 7
1x 7 1x 7
x 49 141x 7
(x 7)
x 56 x 141x 7
56 14 1x 7
4 1x 7
16 x 7
x9
p. 618
© 2001 McGraw-Hill Companies
x 24
SUMMARY
DEFINITION /PROCEDURE
EXAMPLE
REFERENCE
Geometric and Other Applications of Radical Expressions
Pythagorean Theorem
Section 8.5
Given triangle
c
a
659
x
3
7
b
x2 72 32
a2 b2 c2
x2 49 9
In a right triangle, the sum of the squares of the two sides is
equal to the square of the hypotenuse.
x 158
7.6
p. 627
Rational Exponents
Section 8.6
Rational exponents are an alternate way of indicating roots. We
use the following definition.
If a is any real number and n is a positive integer (n 1),
n
a1n 1 a
We restrict a so that a is nonnegative when n is even.
We also define the following.
For any real number a and positive integers m and n, with
n 1, then
n
n
amn (1 a)m 2 am
36
12
136 6
3
2713 127 3
5
24315 1243 3
1
1
125
5
2512 3
2723 (127)2
32 9
4
(a4b8)34 2(a4b8)3
4
2a12b24 a3b6
p. 635
Properties of Exponents
The following five properties for exponents continue to hold
for rational exponents.
Product Rule
am an amn
x12 x13 x1213 x56
p. 638
am
am n
an
x32
x32 12 x22 x
x12
p. 638
(am)n am n
(x13)5 x13 5 x53
p. 638
(ab)m ambm
(2xy)12 212x12y12
p. 638
Quotient Rule
Power Rule
Product-Power Rule
© 2001 McGraw-Hill Companies
Quotient-Power Rule
b
a
m
am
bm
x13
3
2
(x13)2
32
x23
9
Complex Numbers
p. 638
Section 8.7
The number i is defined as
i 11
116 4i
18 2i12
Continued
660
CHAPTER 8
RADICAL EXPRESSIONS
DEFINITION /PROCEDURE
EXAMPLE
REFERENCE
Complex Numbers
Section 8.7
Note that this means that
i2 1
A complex number is any number that can be written in
the form
a bi
in which a and b are real numbers and
i 11
Addition and Subtraction
For the complex numbers a bi and c di,
(a bi) (c di) (a c) (b d )i
p. 647
(2 3i) (3 5i)
(2 3) (3 5)i
1 2i
and
(a bi) (c di) (a c) (b d )i
(5 2i) (3 4i)
(5 3) (2 (4))i
2 2i
(a bi)(c di) (ac bd) (ad bc)i
Note: It is generally easier to use the FOIL multiplication
pattern and the definition of i, rather than to apply the above
formula.
Division
To divide two complex numbers, we multiply the numerator
and denominator by the complex conjugate of the denominator
and write the result in standard form.
(2 5i)(3 4i)
6 8i 15i 20i2
6 7i 20(1)
26 7i
p. 649
3 2i
(3 2i)(3 2i)
3 2i
(3 2i)(3 2i)
9 6i 6i 4i2
9 4i2
9 12i 4(1)
9 4(1)
5
12
5 12i
i
13
13
13
p. 651
© 2001 McGraw-Hill Companies
Multiplication
For the complex numbers a bi and c di,
p. 648
Summary Exercises
This summary exercise set is provided to give you practice with each of the objectives of the chapter. Each exercise is
keyed to the appropriate chapter section. The answers are provided in the instructor’s manual that accompanies this text.
Your instructor will provide guidelines on how to best use these exercises in your instructional program.
[8.1]
Evaluate each of the following roots over the set of real numbers.
1. 1121
2. 164
3
5. 164
6. 181
3
8
A 27
2
9. 28
4. 164
7.
9
A 16
3
8.
3. 181
4
Simplify each of the following expressions. Assume that all variables represent positive real numbers for all subsequent
exercises in this exercise set.
10. 24x2
11. 2a4
13. 249w4z6
14. 2x9
3
16. 28r3s9
[8.2]
12. 236y2
3
15. 227b6
4
18. 232p5q15
17. 216x4y8
3
5
Use the product property to write each of the following expressions in simplified form.
19. 145
20. 175
22. 2108a3
23. 132
3
21. 260x2
3
24. 280w4z3
© 2001 McGraw-Hill Companies
Use the quotient property to write each of the following expressions in simplified form.
25.
9
A 16
26.
7
A 36
27.
y4
B 49
28.
2x
A9
29.
5
A 16x2
30.
5a2
B 27
[8.3]
3
Simplify each of the following expressions if necessary. Then add or subtract as indicated.
3
3
31. 7110 4110
32. 513x 213x
33. 712x 3 12x
34. 8110 3110 2110
35. 172 150
36. 154 124
37. 917 2163
38. 120 145 21125
39. 2116 3154
3
3
661
662
CHAPTER 8
RADICAL EXPRESSIONS
3
40. 227w w112w
43. 172x [8.3]
x
A2
3
3
5
2
41. 2128a 6a22a
3
42. 120 3 a
A9
4
44. 281a a
45.
3
15
115
1
3
115
Multiply and simplify each of the following expressions.
3
3
46. 13x 17y
2
47. 26x 118
2
2
48. 24a b 2ab
49. 15(13 2)
50. 16(18 12)
51. 1a(15a 1125a)
52. (13 5)(13 7)
53. (17 12)(17 13)
54. (15 2)(15 2)
55. (17 13)(17 13)
56. (2 13)2
57. (15 12)2
Rationalize the denominator, and write each of the following expressions in simplified form.
58.
3
A7
59.
3
61.
A a2
112
1x
60.
3
2
62. 3
13x
3
110a
15b
63.
2x2
3
2y5
Divide and simplify each of the following expressions.
64.
1
3 12
65.
11
5 13
66.
15 2
15 2
67.
1x 3
1x 3
Solve each of the following equations. Be sure to check your solutions.
68. 1x 5 4
69. 13x 2 2 5
70. 1y 7 y 5
71. 12x 1 x 8
3
3
72. 15x 2 3
2
73. 2x 2 3 0
74. 1z 7 1 1z
75. 14x 5 1x 1 3
Solve each of the following equations for the indicated variable.
2
2
76. r 2x y
for x
77. t 2p
l
A 10
for l
© 2001 McGraw-Hill Companies
[8.4]
SUMMARY EXERCISES
[8.5]
663
Use the Pythagorean theorem to find the length x in each triangle. Write answers to the nearest tenth.
78.
79.
x
14 in.
10 cm
x
8 in.
9 cm
80. Find the width of a rectangle whose diagonal is 12 ft and whose length is 6 ft.
81. If a window is 3 ft on its shorter side, how high should it be, to the nearest tenth of a foot, to be a golden
rectangle?
82. The base of a ladder that is 16 ft long is placed 4 ft from a wall of a building. How high up the wall is the top of the
ladder?
83. How long must a ladder be to reach 8 m up the side of a building when the base of the ladder is 2 m from the
building?
[8.6]
Evaluate each of the following expressions.
84. 4912
85. 10012
86. (27)13
87. 1614
88. 6423
89. 2532
91. 4912
92. 8134
90.
9
4
32
Use the properties of exponents to simplify each of the following expressions.
93. x32 x52
© 2001 McGraw-Hill Companies
95.
94. b23 b32
r85
r35
96.
98. (y43)6
97. (x35)23
100. (16x13 y23)34
99. (x45y32)10
101.
x2y16
x4y
a54
a12
3
102.
27y3z6
x3
13
664
CHAPTER 8
RADICAL EXPRESSIONS
Write each of the following expressions in radical form.
103. x34
104. (w2z)25
105. 3a23
106. (3a)23
Write each of the following expressions, using rational exponents, and simplify when necessary.
5
4
108. 216w
3
110. 216a8b16
107. 17x
109. 227p3q9
[8.7]
4
Write each of the following roots as a multiple of i. Simplify your result.
111. 149
112. 113
113. 160
Perform the indicated operations.
114. (2 3i) (3 5i)
115. (7 3i) (3 2i)
116. (5 3i) (2 5i)
117. (4 2i) (1 3i)
Find each of the following products.
118. 4i(7 2i)
119. (5 2i)(3 4i)
120. (3 4i)2
121. (2 3i)(2 3i)
122.
5 15i
5i
123.
10
3 4i
124.
3 2i
3 2i
125.
5 10i
2i
© 2001 McGraw-Hill Companies
Find each of the following quotients, and write your answer in standard form.