complex numbers class work
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Pre-Calc Polar & Complex #s ~1~ NJCTL.org
Complex Numbers β Class Work
Simplify using i.
1. ββ16 2. ββ36π4 3. ββ8π2
4. ββ32π₯6π¦7 5. ββ16 β ββ25 6. ββ8 β ββ10
7. 3π β 4π β 5π 8. β2π β 4π β β6π β 8π 9. π9
10. π22 11. π75
Complex Numbers β Homework
Simplify using i.
12. ββ81 13. ββ121π8 14. ββ18π6
15. ββ48π₯5π¦6 16. ββ9 β ββ4 17. ββ12 β ββ75
18. 2π β 5π β 7π 19. βπ β β3π β β5π β β7π 20. π10
21. π23 22. π72
Pre-Calc Polar & Complex #s ~2~ NJCTL.org
Adding, Subtracting, and Multiplying Complex Numbers β Class Work
Simplify
23. (6 + 5π) + (4 + 3π) 24. (7 + 4π) + (β2 β 2π)
25. (β3 β 2π) + (3 β π) 26. (6 + 5π) β (4 + 3π)
27. (7 + 4π) β (β2 β 2π) 28. (β3 β 2π) β (3 β π)
29. 5(4 β 2π) 30. 2π(β6 + π)
31. (6 + 5π)(4 + 3π) 32. (7 + 4π)(β2 β 2π)
33. (β3 β 2π)(3 β π) 34. (8 β 3π)(1 β π)
35. (4 β 2π)2 36. (β6 + π)2
Pre-Calc Polar & Complex #s ~3~ NJCTL.org
Adding, Subtracting, and Multiplying Complex Numbers β Homework
Simplify
37. (2 + 3π) + (8 + 2π) 38. (4 + 9π) + (β4 β 9π)
39. (10 β 7π) + (5 β 3π) 40. (2 + 3π) β (8 + 2π)
41. (4 + 9π) β (β4 β 9π) 42. (10 β 7π) β (5 β 3π)
43. 6(5 β 6π) 44. 2π(4 β 3π)
45. (2 + 3π)(8 + 2π) 46. (4 + 9π)(β4 β 9π)
47. (10 β 7π)(5 β 3π) 48. (β6 β π)(2 β 7π)
49. (6 β 3π)2 50. (β7 + 2π)2
Pre-Calc Polar & Complex #s ~4~ NJCTL.org
Dividing Complex Numbers β Class Work
Simplify
51. 2
π 52.
3
4π 53.
β2
3π
54. 2+π
π 55.
2
1+π 56.
3
2βπ
57. 2+π
3+π 58.
4βπ
3β2π
Dividing Complex Numbers β Homework
Simplify
59. 3
π 60.
2
5π 61.
β4
7π
62. 4βπ
π 63.
8
3+π 64.
2π
4βπ
65. 2βπ
2+3π 66.
5βπ
4β3π
Pre-Calc Polar & Complex #s ~5~ NJCTL.org
Graphing Complex Numbers β Class Work
Determine the quadrant of each of the following.
67. 9 β 3i 68. -2 + 4i
69. (5 + 4i) β (6 β 3i) 70. -3i(4 β 5i)
71. (2 + 3i)2 72. 3βi
i
73. 2
4+i 74.
5β3i
2+4i
Homework
Determine the quadrant of each of the following.
75. -7 β 3i 76. 5 - 4i
77. (3 + 2i) β (-5 + 4i) 78. (3 β i)(-4 + 5i)
79. (-1 + 5i)2 80. β2βi
3i
81. 4
3βi 82.
β6+2i
3β2i
Pre-Calc Polar & Complex #s ~6~ NJCTL.org
Polar Properties β Class Work
Name the point three other ways using polar coordinates.
83. [5,Ο
2] 84. [β4,
2Ο
3]
85. [3,β4Ο
7] 86. [β6,0]
Convert the point to rectangular form.
87. [5,Ο
2] 88. [β4,
2Ο
3]
89. [3,β4Ο
7] 90. [β6,0]
Convert the point to polar form.
91. ( 3, 6) 92. (-4, 2)
93. (1, 0) 94. (7, 7)
Pre-Calc Polar & Complex #s ~7~ NJCTL.org
Polar Properties β Homework
Name the point three other ways using polar coordinates.
95. [7,Ο
3] 96. [β6,
2Ο
5]
97. [2,β3Ο
5] 98. [3, π]
Convert the point to rectangular form.
99. [7,Ο
3] 100. [β6,
2Ο
5]
101. [2,β3Ο
5] 102. [3, Ο]
Convert the point to polar form.
103. ( -3, 2) 104. (-7, -8)
105. (5, 10) 106. (-7, 0)
Pre-Calc Polar & Complex #s ~8~ NJCTL.org
Geometry of Complex Numbers β Class Work
Let a =3 + 4i and b= -2 + 5i, perform the operation and write the answer in complex, rectangular,
polar, and trigonometric forms.
107. a + b 108. b β a
109. ab 110. a2
111. b2 112. 3a2b
113. π = 4(πππ π
4+ ππ ππ
π
4) and π = 3(πππ
7π
6+ ππ ππ
7π
6), find ab.
114. π = [5,2π
5] and π = [3,
4π
6], find cd. 115. Find z if z[10, 80Β°]= [15, 140Β°]
Pre-Calc Polar & Complex #s ~9~ NJCTL.org
Geometry of Complex Numbers β Homework
Let a =7 - 3i and b= -3 - 8i, perform the operation and write the answer in complex, rectangular,
polar, and trigonometric forms.
116. a + b 117. a β b
118. b β a 119. ab
120. a2 121. b2
122. 3a 123. 3a2b
124. π = 7(πππ π
3+ ππ ππ
π
3) and π = 2(πππ
5π
6+ ππ ππ
5π
6), find ab.
125. π = [12,7π
4] and π = [. 5,
5π
3], find cd. 126. Find z if z[20, 100Β°]= [15, 140Β°]
Pre-Calc Polar & Complex #s ~10~ NJCTL.org
Polar Equations and Graphs β Class Work
127. Draw the graph of π = sin π 128. Draw the graph of π = 3 + πππ π
129. Draw the graph of π = 5 130. Draw the graph of π =2π
3
131. Draw the graph of ππππ π = 6
Pre-Calc Polar & Complex #s ~11~ NJCTL.org
Polar Equations and Graphs β Homework
132. Draw the graph of π = πππ π 133. Draw the graph of π = 4 + π πππ
134. Draw the graph of π = β5 135. Draw the graph of π =3π
4
136. Draw the graph of ππ πππ = β6
Pre-Calc Polar & Complex #s ~12~ NJCTL.org
Rose Curves and Spirals β Class Work
137. How many petals and what is a petals length for π = 4πππ 3π? Draw the graph.
138. How many petals and what is a petals length for π = 5π ππ6π? Draw the graph.
139. How many petals and what is a petals length for π = 2πππ 4π? Draw the graph.
140. How many petals and what is a petals length for π = 7πππ 5π? Draw the graph.
141. What kind of spiral is π = 3π? 142. What kind of spiral is π = 2π + 2?
Pre-Calc Polar & Complex #s ~13~ NJCTL.org
Rose Curves and Spirals β Homework
143. How many petals and what is a petals length for π = 6πππ 2π? Draw the graph.
144. How many petals and what is a petals length for π = 4π ππ7π? Draw the graph.
145. How many petals and what is a petals length for π = 3πππ 6π? Draw the graph.
146. How many petals and what is a petals length for π = 5πππ 3π? Draw the graph.
147. What kind of spiral is π = 2π? 148. What kind of spiral is π = 3π + 1?
Pre-Calc Polar & Complex #s ~14~ NJCTL.org
Powers of Complex Numbers β Class Work
Compute the given power and write your answer in the original form.
149. ([3,60Β°])5 150. (4 (πππ π
5+ ππ ππ
π
5))
7
151. (5 β 6π)6 152. (β5,9)8
153. If a tenth root of w is (3,8) what is w?
Homework
Compute the given power and write your answer in the original form.
154. ([9,80Β°])7 155. (5 (πππ 4π
3+ ππ ππ
4π
3))
9
156. (β4 + 7π)8 157. (β7, β3)10
158. If a sixth root of w is 7(πππ 0 + ππ ππ0) what is w?
Pre-Calc Polar & Complex #s ~15~ NJCTL.org
Roots of Complex Numbers β Class Work
Find the given roots and write the answer in the same form as the original.
159. fifth root of [3,60Β°] 160. fourth root of 4 (πππ π
5+ ππ ππ
π
5)
161. sixth root of 5 β 6π 162. eighth root of (β5,9)
163. a to the fourth is β3(cos 20Β° + ππ ππ 20Β°), find a
Pre-Calc Polar & Complex #s ~16~ NJCTL.org
Homework
Find the given roots and write the answer in the same form as the original.
164. fifth root of [9,80Β°] 165. fourth root of 5 (πππ 4π
3+ ππ ππ
4π
3)
166. sixth root of (β4 + 7π) 167. eighth root of (β7, β3)
168. a to the sixth is β3(cos 30Β° + ππ ππ 30Β°), find a
Pre-Calc Polar & Complex #s ~17~ NJCTL.org
Polar and Complex Numbers Unit Review
Multiple Choice
1. Simplify: β4π β 6π β β2π β βπ
a. -48i
b. 48i
c. -48
d. 48
2. Simplify: (6 β π)2
a. 35 + 12i
b. 35 - 12i
c. 37 - 12i
d. 37 + 12i
3. Simplify: 3βπ
4β2π
a. 7
10+
1
10i
b. 7
6+
1
6i
c. 7
10β
1
10i
d. 7
6β
1
6i
4. What quadrant is (6 + 2i) β (7 β 4i) in?
a. I
b. II
c. III
d. IV
5. What quadrant is (3 - 5i)2 in?
a. I
b. II
c. III
d. IV
6. What quadrant is 3βπ
4β2π in?
a. I
b. II
c. III
d. IV
7. Which of the point choices listed are not equal to: [5,Ο
2]
a. (0,5)
b. 5(πππ π
2+ ππ ππ
π
2)
c. [β5,3Ο
2]
d. they are all equivalent
Pre-Calc Polar & Complex #s ~18~ NJCTL.org
8. Convert the point to rectangular form: [4,Ο
3]
a. (2,β3
2)
b. (β3
2, 2)
c. (2,β3)
d. (2,2β3)
9. Convert the point to polar form: ( 2.5 , 6)
a. (6.5, 0.395)
b. (6.5 , 1.176Β°)
c. (6.5 , 22.620Β°)
d. (6.5, 67.380Β°)
10. Let a =8 - 2i and b= -5 - 7i, which of the following is not a + b?
a. (3,-9)
b. [3β10, β71.565]
c. 10(cos 288.435Β° + i sin 288.435Β°)
d. β( -3 + 9i)
11. π = 6(πππ π
4+ ππ ππ
π
4) and π = β3(πππ
5π
3+ ππ ππ
5π
3), find ab.
a. β18(πππ 6π
7+ ππ ππ
6π
7)
b. β18(πππ 5π
12+ ππ ππ
5π
12)
c. β18(πππ 17π
12+ ππ ππ
17π
12)
d. β18(πππ 23π
12+ ππ ππ
23π
12)
12. How many petals and what is a petals length for π = 4πππ 8π?
a. 4 petals, length 8
b. 8 petals, length 4
c. 8 petals, length 8
d. 16 ptdals, length 4
13. Compute: (7 β 3π)6
a. ( 195112, 220.809Β°)
b. ( 45.694, 220.809Β°)
c. ( 195112, 1.871π)
d. ( 45.694, 1.871π)
14. If a tenth root of w is [5,2π
3], what is w?
a. [50,20Ο
3]
b. [9765625,20Ο
3]
c. [50,4Ο
3]
d. [9765625,4Ο
3]
Pre-Calc Polar & Complex #s ~19~ NJCTL.org
15. Find the third root of 27 (πππ π
2β ππ ππ
π
2)
a. [3,Ο
6]
b. [3,Ο+4kΟ
6] for k β {1,2}
c. [3,4+kΟ
6] for k β {1,2,3}
d. [3,Ο+4kΟ
6] for k β {0,1,2}
Extended Response
16. Let a =8 - 2i and b= -5 - 7i.
a. Find 3a2b.
b. How far from the origin is a + b?
c. What is the angle of rotation of a+b?
17. Write an equation
a. for a rose curve with 8 petals of length 5
b. for a rose curve with 5 petals of length 6
c. a Spiral of Archimedes with 6π between the spirals