8.3.1: Polar Form of Complex Numbers (Exercise)
section 8.3 exercises
Simplify each expression to a single complex number.
1. \(\sqrt{-9}\)
2. \(\sqrt{-16}\)
3. \(\sqrt{-6} \sqrt{-24}\)
4. \(\sqrt{-3} \sqrt{-75}\)
5. \(\dfrac{2 + \sqrt{-12}}{2}\)
6. \(\dfrac{4 + \sqrt{-20}}{20}\)
Simplify each expression to a single complex number.
7. \((3 + 2i) + (5 - 3i)\)
8. \((-2 - 4i) + (1 + 6i)\)
9. \(-5 + 3i) - (6 - i)\)
10. \((2 - 3i) - (3 + 2i)\)
11. \((2 + 3i) (4i)\)
12. \((5 - 2i)(3i)\)
13. \(6 - 2i)(5)\)
14. \(-2 + 4i) (8)\)
15. \((2 + 3i) (4 - i)\)
16. \((-1 + 2i)(-2 + 3i)\)
17. \((4 - 2i) (4 + 2i)\)
18. \((3 + 4i)(3 - 4i)\)
19. \(\dfrac{3+ 4i}{2}\)
20. \(\dfrac{6- 2i}{3}\)
21. \(\dfrac{-5 + 3i}{2i}\)
22. \(\dfrac{6 + 4i}{i}\)
23. \(\dfrac{2 - 3i}{4 + 3i}\)
24. \(\dfrac{3 + 4i}{2 - i}\)
25. \(i^6\)
26. \(i^{11}\)
27. \(i^{17}\)
28. \(i^{24}\)
Rewrite each complex number from polar form into \(a + bi\) form.
29. \(3e^{2i}\)
30. \(4e^{4i}\)
31. \(6e^{\dfrac{\pi}{6}i}\)
32. \(8e^{\dfrac{\pi}{3}i}\)
33. \(3e^{\dfrac{5\pi}{4}i}\)
34. \(5e^{\dfrac{7\pi}{4}i}\)
Rewrite each complex number into polar \(re^{i\theta}\) form.
35. 6
36. -8
37. \(-4i\)
38. \(6i\)
39. \(2+ 2i\)
40. \(4 + 4i\)
41. \(-3 + 3i\)
42. \(-4 - 4i\)
43. \(5 + 3i\)
44. \(4 + 7i\)
45. \(-3 + i\)
46. \(-2 + 3i\)
47. \(-1 - 4i\)
48. \(-3 - 6i\)
49. \(5 - i\)
50. \(1- 3i\)
Compute each of the following, leaving the result in polar \(re^{i\theta}\) form.
51. \((3e^{\dfrac{\pi}{6}i})(2e^{\dfrac{\pi}{4}i})\)
52. \((2e^{\dfrac{2\pi}{3}i})(4e^{\dfrac{5\pi}{3}i})\)
53. \(\dfrac{6e^{\dfrac{3\pi}{4}i}}{3e^{\dfrac{\pi}{6} i}}\)
54. \(\dfrac{24e^{\dfrac{4\pi}{3}i}}{6e^{\dfrac{\pi}{2} i}}\)
55. \((2 e^{\dfrac{\pi}{4}i})^{10}\)
56. \((3 e^{\dfrac{\pi}{6}i})^{4}\)
57. \(\sqrt{16e^{\dfrac{2\pi}{3}i}}\)
58. \(\sqrt{9e^{\dfrac{3\pi}{2}i}}\)
Compute each of the following, simplifying the result into \(a + bi\) form.
59. \((2 + 2i)^8\)
60. \((4 + 4i)^6\)
61. \(\sqrt{-3 + 3i}\)
62. \(\sqrt{-4 - 4i}\)
63. \(\sqrt[3]{5 + 3i}\)
64. \(\sqrt[4]{4 + 7i}\)
Solve each of the following equations for all complex solutions.
65. \(z^5 = 2\)
66. \(z^7 = 3\)
67. \(z^6 = 1\)
68. \(z^8 = 1\)
- Answer
-
1. \(3i\)
3. -12
5. \(1 + \sqrt{3}i\)
7. \(8 - i\)
9. \(-11 + 4i\)
11. \(-12 + 8i\)
13. \(30 - 10i\)
15. \(11 + 10i\)
17. 20
19. \(\dfrac{3}{2} + 2i\)
21. \(\dfrac{3}{2} + \dfrac{5}{2} i\)
23. \(-\dfrac{1}{25} - \dfrac{18}{25}i\)
25. -1
27. \(i\)
29. \(3\cos(2) + 3\sin(2) i = -1.248 + 2.728i\)
31. \(3\sqrt{3} + 3i\)
33. \(-\dfrac{3\sqrt{2}}{2} - \dfrac{3\sqrt{2}}{2} i\)
35. \(6e^(0i)\)
37. \(4e^(\dfrac{3\pi}{2}i)\)
39. \(2\sqrt{2} e^(\dfrac{\pi}{4}i)\)
41. \(3\sqrt{2} e^(\dfrac{3\pi}{4}i)\)
43. \(\sqrt{34}e^(0.540i)\)
45. \(\sqrt{10}e^(2.820i)\)
47. \(\sqrt{17}e^(4.467i)\)
49. \(\sqrt{26}e^(6.086i)\)
51. \(6e^(\dfrac{5\pi}{12}i)\)
53. \(2e^(\dfrac{7\pi}{12}i)\)
55. \(1024e^(\dfrac{5\pi}{2}i)\)
57. \(4e^(\dfrac{\pi}{3}i)\)
59. 4096
61. \(0.788 + 1.903i\)
63. \(1.771 + 0.322i\)
65. \(\sqrt[5]{2} \approx 1.149, 0.355 + 1.092i, -0.929 + 0.675i, -0.929 - 0.675i, 0.355 - 1.092i\)
67. \(1, \dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i, -\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i, -1, -\dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i, \dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i\)