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8.9E: Exercises - Mathematics


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. (isqrt{11}) c. (2isqrt{2})

3. (10i) b. (isqrt{13}) c. (3isqrt{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(9sqrt{3} ight)i)

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

9. (8+7i)

11. (14+2i)

13. (-2+2i)

15. (8+5i)

17. (7-13i)

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

39. (-20-left(2 sqrt{2} ight) 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} ight)^{10}cdot i= i)

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

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

67. (i^{137} = i^{136}cdot i = left(i^{4} ight)^{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?
(egin{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.

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?


Watch the video: Unit 2: Organizing and Analyzing Data (October 2021).