# 8.9E: Exercises

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### Practice Makes Perfect

##### Exercises 1 - 4: Evaluate a Square Root of a Negative Number

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. $$\sqrt{-75}$$

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

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

##### ExerciseS 5 - 21: Add or Subtract Complex Numbers

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

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

##### ExerciseS 21 - 28: Multiply Complex Numbers

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. $$(-6-2 i)(-3-5 i)$$

21. $$12+20i$$

23. $$-12+18i$$

25. $$-38+9 i$$

27. $$27+15i$$

##### ExerciseS 29 - 32: Multiply Complex Numbers

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. $$(-6-5 i)^{2}$$

29. $$-7+24i$$

31. $$-5-12i$$

##### Exercises 33 - 46: Multiply Complex Numbers

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. $$(-3-8 i)(-3+8 i)$$

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. $$53 = 53 + 0i$$

##### ExerciseS 45 - 49: Multiply Complex Numbers

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. $$(-3-4 i)(-3+4 i)$$

45. $$50$$

47. $$85$$

##### ExerciseS 49 - 60: Divide Complex Numbers

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. $$\dfrac{-3-5 i}{2 i}$$

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. $$-\frac{3}{4}+\frac{1}{2} i$$

##### ExerciseS 61 - 68: Simplify Powers of $$i$$

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. $$i^{255}$$

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

##### ExerciseS 69 - 72: Writing Exercises

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.