
# 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. $$z^8 = 1$$

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. $$1, \dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i, -\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i, -1, -\dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i, \dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i$$