College Algebra 9th txtbk

SECTION 1–4 Complex Numbers 83
36. (5 2i)(4 3i) 37. (2 9i)(2 9i)
38. (3 8i)(3 8i) 39.
i
4 3i
40. 41. 42.
3 i 1 2i
7 i
43. 44.
2 i
In Problems 45–52, evaluate and express results in standard form.
45. 12 18
46. 13 112
47. 12 18
48. 13 112
49. 12 18
50. 13 112
51. 12 18
52. 13 112
In Problems 53–62, convert imaginary numbers to standard form,
perform the indicated operations, and express answers in standard
form.
53. (2 14) (5 19)
54. (3 14) (8 125)
55. (9 19) (12 125)
56. (2 136) (4 149)
57. (3 14)(2 149)
58. (2 11)(5 19)
5 14
6 164
59. 60.
7
2
1
1
61. 62.
2 19
3 116
In Problems 63–68, write the complex number in standard form.
1
63. 5 64.
i
10i
65. (2i) 2 5(2i) 6 66. (i13) 4 2(i13) 2 15
67. (5 2i) 2 4(5 2i) 1
68. (7 3i) 2 8(7 3i) 30
69. Evaluate x 2 2x 2 for x 1 i.
70. Evaluate x 2 2x 2 for x 1 i.
In Problems 71–74, for what real values of x does each expression
represent an imaginary number?
71. 13 x
72. 15 x
73. 12 3x
74. 13 2x
In Problems 75–78, solve for x and y.
75. (2x 1) (3y 2)i 5 4i
1
2 4i
3 5i
2 i
5 10i
3 4i
76. 3x ( y 2)i (5 2x) (3y 8)i
(1 x) ( y 2)i
77.
2 i
1 i
(2 x) ( y 3)i
78.
3 i
1 i
In Problems 79–82, solve for z and write your answer in standard
form.
79. (10 2i)z (5 i) 2i
80. (3 2i)z (4i 6) 8i
81. (4 2i)z (7 2i) (4 i)z (3 5i)
82. (2 3i) (4 5i)z (1 i) (4 2i)z
83. Show that 2 i and 2 i are square roots of 3 4i.
84. Show that 3 2i and 3 2i are square roots of 5 12i.
85. Explain what is wrong with the following “proof ” that
1 1:
1 i 2 11 11 1(1)(1) 11 1
86. Explain what is wrong with the following “proof ” that 1i i.
What is the correct value of 1i?
1
i 1
11 11
11 1
A 1 11 i
87. Show that i 4k 1, k a natural number
88. Show that i 4k1 i, k a natural number
Supply the reasons in the proofs for the theorems stated in
Problems 89 and 90.
89. Theorem: The complex numbers are commutative under
addition.
Proof: Let a bi and c di be two arbitrary complex
numbers; then:
Statement
1. (a bi) (c di) (a c) (b d )i
2.
(c a) (d b)i
3.
(c di) (a bi)
Reason
1.
2.
3.
90. Theorem: The complex numbers are commutative under
multiplication.
Proof: Let a bi and c di be two arbitrary complex
numbers; then:
Statement
1. (a bi) (c di) (ac bd ) (ad bc)i
2.
(ca db) (da cb)i
3.
(c di)(a bi)
Reason
1.
2.
3.