8.9E: Exercises
- Last updated
- Feb 14, 2022
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Practice Makes Perfect
In the following exercises, write each expression in terms of i and simplify if possible.
- a. √−16 b. √−11 c. √−8
- a. √−121 b. √−1 c. √−20
- a. √−100 b. √−13 c. √−45
- a. √−49 b. √−15 c. √−75
- Answer
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1. a. 4i b. i√11 c. 2i√2
3. a. 10i b. i√13 c. 3i√5
In the following exercises, add or subtract, putting the answer in a+bi form.
5. √−75+√−48
6. √−12+√−75
7. √−50+√−18
8. √−72+√−8
9. (1+3i)+(7+4i)
10. (6+2i)+(3−4i)
11. (8−i)+(6+3i)
12. (7−4i)+(−2−6i)
13. (1−4i)−(3−6i)
14. (8−4i)−(3+7i)
15. (6+i)−(−2−4i)
16. (−2+5i)−(−5+6i)
17. (5−√−36)+(2−√−49)
18. (−3+√−64)+(5−√−16)
19. (−7−√−50)−(−32−√−18)
20. (−5+√−27)−(−4−√−48)
- Answer
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5. 0+(9√3)i
7. 0+(8√2)i
9. 8+7i
11. 14+2i
13. −2+2i
15. 8+5i
17. 7−13i
19. 25−(2√2)i
In the following exercises, multiply, putting the answer in a+bi form.
21. 4i(5−3i)
22. 2i(−3+4i)
23. −6i(−3−2i)
24. −i(6+5i)
25. (4+3i)(−5+6i)
26. (−2−5i)(−4+3i)
27. (−3+3i)(−2−7i)
28. (−6−2i)(−3−5i)
- Answer
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21. 12+20i
23. −12+18i
25. −38+9i
27. 27+15i
In the following exercises, multiply using the Product of Binomial Squares Pattern, putting the answer in a+bi form.
29. (3+4i)2
30. (−1+5i)2
31. (−2−3i)2
32. (−6−5i)2
- Answer
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29. −7+24i
31. −5−12i
In the following exercises, multiply, putting the answer in a+bi form.
33. √−25⋅√−36
34. √−4⋅√−16
35. √−9⋅√−100
36. √−64⋅√−9
37. (−2−√−27)(4−√−48)
38. (5−√−12)(−3+√−75)
39. (2+√−8)(−4+√−18)
40. (5+√−18)(−2−√−50)
41. (2−i)(2+i)
42. (4−5i)(4+5i)
43. (7−2i)(7+2i)
44. (−3−8i)(−3+8i)
- Answer
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33. 30i=0+30i
35. −30=−30+0i
37. −44+(4√3)i
39. −20−(2√2)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. (6−5i)(6+5i)
47. (9−2i)(9+2i)
48. (−3−4i)(−3+4i)
- Answer
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45. 50
47. 85
In the following exercises, divide, putting the answer in a+bi form.
49. 3+4i4−3i
50. 5−2i2+5i
51. 2+i3−4i
52. 3−2i6+i
53. 32−3i
54. 24−5i
55. −43−2i
56. −13+2i
57. 1+4i3i
58. 4+3i7i
59. −2−3i4i
60. −3−5i2i
- Answer
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49. i=0+i
51. 225+1125i
53. 613+913i
55. −1213−813i
57. 43−13i
59. −34+12i
In the following exercises, simplify.
61. i41
62. i39
63. i66
64. i48
65. i128
66. i162
67. i137
68. i255
- Answer
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61. i41=i40⋅i=(i4)10⋅i=i
63. i66=i64⋅i2=(i4)16⋅(−1)=−1
65. i128=(i4)32=1
67. i137=i136⋅i=(i4)34⋅i=1⋅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?
√−7⋅√−7√497
71. Why is √−64=8i but 3√−64=−4.
72. Explain how dividing complex numbers is similar to rationalizing a denominator.
- Answer
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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?
