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Mathematics LibreTexts

8.8E: Exercises

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    30903
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    Practice Makes Perfect

    In the following exercises, write each expression in terms of \(i\) and simplify if possible.

    1. a. \(\sqrt{-16}\) b. \(\sqrt{-11}\) c. \(\sqrt{-8}\)
    2. a. \(\sqrt{-121}\) b. \(\sqrt{-1}\) c. \(\sqrt{-20}\)
    3. a. \(\sqrt{-100}\) b. \(\sqrt{-13}\) c. \(\sqrt{-45}\)
    4. a. \(\sqrt{-49}\) b. \(\sqrt{-15}\) c. \(\sqrt{-75}\)
    Answer

    1. a. \(4i\) b. \(i\sqrt{11}\) c. \(2i\sqrt{2}\)

    3. a. \(10i\) b. \(i\sqrt{13}\) c. \(3i\sqrt{5}\)

    In the following exercises, add or subtract, putting the answer in \(a + bi\) form.

    5. \(\sqrt{-75}+\sqrt{-48}\)

    6. \(\sqrt{-12}+\sqrt{-75}\)

    7. \(\sqrt{-50}+\sqrt{-18}\)

    8. \(\sqrt{-72}+\sqrt{-8}\)

    9. \((1+3 i)+(7+4 i)\)

    10. \((6+2 i)+(3-4 i)\)

    11. \((8-i)+(6+3 i)\)

    12. \((7-4 i)+(-2-6 i)\)

    13. \((1-4 i)-(3-6 i)\)

    14. \((8-4 i)-(3+7 i)\)

    15. \((6+i)-(-2-4 i)\)

    16. \((-2+5 i)-(-5+6 i)\)

    17. \((5-\sqrt{-36})+(2-\sqrt{-49})\)

    18. \((-3+\sqrt{-64})+(5-\sqrt{-16})\)

    19. \((-7-\sqrt{-50})-(-32-\sqrt{-18})\)

    20. \((-5+\sqrt{-27})-(-4-\sqrt{-48})\)

    Answer

    5. \(0+\left(9\sqrt{3}\right)i\)

    7. \(0+\left(8\sqrt{2}\right)i\)

    9. \(8+7i\)

    11. \(14+2i\)

    13. \(-2+2i\)

    15. \(8+5i\)

    17. \(7-13i\)

    19. \(25-\left(2 \sqrt{2}\right) i\)

    In the following exercises, multiply, putting the answer in \(a+bi\) form.

    21. \(4 i(5-3 i)\)

    22. \(2 i(-3+4 i)\)

    23. \(-6 i(-3-2 i)\)

    24. \(-i(6+5 i)\)

    25. \((4+3 i)(-5+6 i)\)

    26. \((-2-5 i)(-4+3 i)\)

    27. \((-3+3 i)(-2-7 i)\)

    28. \((-6-2 i)(-3-5 i)\)

    Answer

    21. \(12+20i\)

    23. \(-12+18i\)

    25. \(-38+9 i\)

    27. \(27+15i\)

    In the following exercises, multiply using the Product of Binomial Squares Pattern, putting the answer in \(a+bi\) form.

    29. \((3+4 i)^{2}\)

    30. \((-1+5 i)^{2}\)

    31. \((-2-3 i)^{2}\)

    32. \((-6-5 i)^{2}\)

    Answer

    29. \(-7+24i\)

    31. \(-5-12i\)

    In the following exercises, multiply, putting the answer in \(a+bi\) form.

    33. \(\sqrt{-25} \cdot \sqrt{-36}\)

    34. \(\sqrt{-4} \cdot \sqrt{-16}\)

    35. \(\sqrt{-9} \cdot \sqrt{-100}\)

    36. \(\sqrt{-64} \cdot \sqrt{-9}\)

    37. \((-2-\sqrt{-27})(4-\sqrt{-48})\)

    38. \((5-\sqrt{-12})(-3+\sqrt{-75})\)

    39. \((2+\sqrt{-8})(-4+\sqrt{-18})\)

    40. \((5+\sqrt{-18})(-2-\sqrt{-50})\)

    41. \((2-i)(2+i)\)

    42. \((4-5 i)(4+5 i)\)

    43. \((7-2 i)(7+2 i)\)

    44. \((-3-8 i)(-3+8 i)\)

    Answer

    33. \(30i = 0 + 30i\)

    35. \(-30 = -30 + 0i\)

    37. \(-44+\left(4 \sqrt{3}\right) i\)

    39. \(-20-\left(2 \sqrt{2}\right) i\)

    41. \(5 = 5 + 0i\)

    43. \(53 = 53 + 0i\)

    In the following exercises, multiply using the Product of Complex Conjugates Pattern.

    45. \((7-i)(7+i)\)

    46. \((6-5 i)(6+5 i)\)

    47. \((9-2 i)(9+2 i)\)

    48. \((-3-4 i)(-3+4 i)\)

    Answer

    45. \(50\)

    47. \(85\)

    In the following exercises, divide, putting the answer in \(a+bi\) form.

    49. \(\dfrac{3+4 i}{4-3 i}\)

    50. \(\dfrac{5-2 i}{2+5 i}\)

    51. \(\dfrac{2+i}{3-4 i}\)

    52. \(\dfrac{3-2 i}{6+i}\)

    53. \(\dfrac{3}{2-3 i}\)

    54. \(\dfrac{2}{4-5 i}\)

    55. \(\dfrac{-4}{3-2 i}\)

    56. \(\dfrac{-1}{3+2 i}\)

    57. \(\dfrac{1+4 i}{3 i}\)

    58. \(\dfrac{4+3 i}{7 i}\)

    59. \(\dfrac{-2-3 i}{4 i}\)

    60. \(\dfrac{-3-5 i}{2 i}\)

    Answer

    49. \(i = 0 + i\)

    51. \(\frac{2}{25}+\frac{11}{25} i\)

    53. \(\frac{6}{13}+\frac{9}{13} i\)

    55. \(-\frac{12}{13}-\frac{8}{13} i\)

    57. \(\frac{4}{3}-\frac{1}{3} i\)

    59. \(-\frac{3}{4}+\frac{1}{2} i\)

    In the following exercises, simplify.

    61. \(i^{41}\)

    62. \(i^{39}\)

    63. \(i^{66}\)

    64. \(i^{48}\)

    65. \(i^{128}\)

    66. \(i^{162}\)

    67. \(i^{137}\)

    68. \(i^{255}\)

    Answer

    61. \(i^{41} = i^{40}\cdot i = \left(i^{4}\right)^{10}\cdot i= i\)

    63. \(i^{66} = i^{64}\cdot i^{2} = \left(i^{4}\right)^{16}\cdot (-1)= -1\)

    65. \(i^{128} = \left(i^{4}\right)^{32} = 1\)

    67. \(i^{137} = i^{136}\cdot i = \left(i^{4}\right)^{34}\cdot i = 1 \cdot i = i\)

    69. Explain the relationship between real numbers and complex numbers.

    70. Aniket multiplied as follows and he got the wrong answer. What is wrong with his reasoning?
    \(\begin{array}{c}{\sqrt{-7} \cdot \sqrt{-7}} \\ {\sqrt{49}} \\ {7}\end{array}\)

    71. Why is \(\sqrt{-64}=8 i\) but \(\sqrt[3]{-64}=-4\).

    72. Explain how dividing complex numbers is similar to rationalizing a denominator.

    Answer

    69. Answers may vary

    71. Answers may vary

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    The table has 4 columns and 4 rows. The first row is a header row with the headers “I can…”, “Confidently”, “With some help.”, and “No – I don’t get it!”. The first column contains the phrases “evaluate the square root of a negative number”, “add or subtract complex numbers”, “multiply complex numbers”, “divide complex numbers”, and “simplify powers of i”. The other columns are left blank so the learner can indicate their level of understanding.
    Figure 8.8.15

    b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?