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Mathematics LibreTexts

8.3.1: Polar Form of Complex Numbers (Exercise)

  • Page ID
    22237
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    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. \(1, \dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i, -\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i, -1, -\dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i, \dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i\)