Answer Key 9.9

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1. \(3+8-4i\)
   \(11-4i\)
2. \(3i-7i\)
   \(-4i\)
3. \(7i-3+2i\)
   \(9i-3\)
4. \(5-6-6i\)
   \(-1-6i\)
5. \(-6i-3-7i\)
   \(-13i-3\)
6. \(-8i-7i-5+3i\)
   \(-12i-5\)
7. \(3-3i-7-8i\)
   \(-4-11i\)
8. \(-4-i+1-5i\)
   \(-3-6i\)
9. \(i-2-3i-6\)
   \(-2i-8\)
10. \(5-4i+8-4i\)
    \(13-8i\)
11. \(-48i^2\)
    \(-48(-1)\)
    \(48\)
12. \(-24i^2\)
    \(-24(-1)\)
    \(24\)
13. \(-40i^2\)
    \(-40(-1)\)
    \(40\)
14. \(-32i^2\)
    \(-32(-1)\)
    \(32\)
15. \(49i^2\)
    \(49(-1)\)
    \(-49\)
16. \(-7i^2(4-3i)\)
    \(-28i^2+21i^3\)
    \(-28(-1)+21(-1)i\)
    \(28-21i\)
17. \(36+60i+25i^2\)
    \(36+60i+25(-1)\)
    \(36+60i-25\)
    \(11+60i\)
18. \(16i^2(-2-8i)\)
    \(32i^2+128i^3\)
    \(32(-1)+128(-1)i\)
    \(-32-128i\)
19. \(\begin{array}{rrrrrl}
    \\ \\ \\ \\ \\
    &56&-&42i&& \\
    &&+&32i&-&24i^2 \\
    \midrule
    &56&-&10i&-&24(-1) \\
    +&24&&&\Longleftarrow & \\
    \midrule
    &80&-&10i&&
    \end{array}\)
20. \(9i^2(4-4i)\)
    \(-36i^2+36i^3\)
    \(-36(-1)+36(-1)i\)
    \(36-36i\)
21. \(\begin{array}{rrrrrl}
    \\ \\ \\ \\ \\
    &-8&+&10i&& \\
    &&+&28i&-&35i^2 \\
    \midrule
    &-8&+&38i&-&35(-1) \\
    +&35&&&\Longleftarrow & \\
    \midrule
    &27&+&38i&&
    \end{array}\)
22. \(-32i+64i+4+12i\)
    \(-28+76i\)
23. \(\begin{array}{rrrrrl}
    \\ \\ \\ \\ \\
    &32&+&24i&& \\
    &&-&16i&-&12i^2 \\
    \midrule
    &32&+&8i&-&12(-1) \\
    +&12&&&\Longleftarrow & \\
    \midrule
    &44&+&8i&&
    \end{array}\)
24. \(-18i+12i^2-28i^2\)
    \(-18i+12(-1)-28(-1)\)
    \(-18i-12+28\)
    \(-18i+16\)
25. \(\begin{array}{rrrrrl}
    \\ \\ \\ \\ \\
    &2&+&10i&& \\
    &&+&i&+&5i^2 \\
    \midrule
    &2&+&11i&+&5(-1) \\
    -&5&&&\Longleftarrow & \\
    \midrule
    &-3&+&11i&&
    \end{array}\)
26. \(\begin{array}{rrrrrl}
    \\ \\ \\ \\ \\
    &-6&+&3i&& \\
    &&+&10i&-&5i^2 \\
    \midrule
    &-6&+&13i&-&5(-1) \\
    +&5&&&\Longleftarrow & \\
    \midrule
    &-1&+&13i&&
    \end{array}\)
27. \(\dfrac{-9+5i}{i}\cdot \dfrac{i}{i}\Rightarrow \dfrac{-9i+5i^2}{i^2}\Rightarrow \dfrac{-9i+5(-1)}{(-1)}\Rightarrow 9i+5\)
28. \(\dfrac{-3+2i}{-3i}\cdot \dfrac{i}{i}\Rightarrow \dfrac{-3i+2i^2}{-3i^2}\Rightarrow \dfrac{-3i+2(-1)}{-3(-1)}\Rightarrow \dfrac{-3i-2}{3}\)
29. \(\dfrac{-10i-9i}{6i}\cdot \dfrac{i}{i}\Rightarrow \dfrac{-10i-9i^2}{6i^2}\Rightarrow \dfrac{-10i-9(-1)}{6(-1)}\Rightarrow \dfrac{-10i+9}{-6}\)
30. \(\dfrac{-4+2i}{3i}\cdot \dfrac{i}{i}\Rightarrow \dfrac{-4i+2i^2}{3i^2}\Rightarrow \dfrac{-4i+2(-1)}{3(-1)}\Rightarrow \dfrac{-4i-2}{-3}\)
31. \(\dfrac{-3-6i}{4i}\cdot \dfrac{i}{i}\Rightarrow \dfrac{-3i-6i^2}{4i^2}\Rightarrow \dfrac{-3i-6(-1)}{4(-1)}\Rightarrow \dfrac{-3i+6}{4}\)
32. \(\dfrac{-5+9i}{9i}\cdot \dfrac{i}{i}\Rightarrow \dfrac{-5i+9i^2}{9i^2}\Rightarrow \dfrac{-5i+9(-1)}{9(-1)}\Rightarrow \dfrac{-5i-9}{-9}\Rightarrow \dfrac{5i+9}{9}\)
33. \(\dfrac{10-i}{-i}\cdot \dfrac{i}{i}\Rightarrow \dfrac{10i-i^2}{-i^2}\Rightarrow \dfrac{10i-(-1)}{-(-1)}\Rightarrow 10i+1\)
34. \(\dfrac{10}{5i}\cdot \dfrac{i}{i}\Rightarrow \dfrac{10i}{5i^2}\Rightarrow \dfrac{10i}{5(-1)}\Rightarrow \dfrac{10i\div -5}{-5\div -5}\Rightarrow -2i\)
35. \(\dfrac{4i}{-10+i}\cdot \dfrac{-10-i}{-10-i}\Rightarrow \dfrac{-40i-4i^2}{(-10)^2-(i)^2}\Rightarrow \dfrac{-40i-4(-1)}{100--1}\Rightarrow \dfrac{-40i+4}{101}\)
36. \(\dfrac{9i}{1-5i}\cdot \dfrac{1+5i}{1+5i}\Rightarrow \dfrac{9i+45i^2}{(1)^2-(5i)^2}\Rightarrow \dfrac{9i+45(-1)}{1-25i^2}\Rightarrow \dfrac{9i-45}{1-25(-1)}\Rightarrow \dfrac{9i-45}{1+25}\Rightarrow \\ \)
    \(\dfrac{9i-45}{26}\)
37. \(\dfrac{8}{7-6i}\cdot \dfrac{7+6i}{7+6i}\Rightarrow \dfrac{56+48i}{7^2-36i^2}\Rightarrow \dfrac{56+48i}{49-36(-1)}\Rightarrow \dfrac{56+48i}{49+36}\Rightarrow \dfrac{56+48i}{85}\)
38. \(\dfrac{4}{4+6i}\cdot \dfrac{4-6i}{4-6i}\Rightarrow \dfrac{16-24i}{16-36i^2}\Rightarrow \dfrac{16-24i}{16+36}\Rightarrow \dfrac{(16-24i)\div 4}{52\div 4} \Rightarrow \dfrac{4-6i}{13}\)
39. \(\dfrac{7}{10-7i}\cdot \dfrac{10+7i}{10+7i}\Rightarrow \dfrac{70+49i}{100-49i^2}\Rightarrow \dfrac{70+49i}{100+49}\Rightarrow \dfrac{70+49i}{149}\)
40. \(\dfrac{9}{-8-6i}\cdot \dfrac{-8+6i}{-8+6i}\Rightarrow \dfrac{-72+54i}{64-36i^2}\Rightarrow \dfrac{-72+54i}{64+36}\Rightarrow \dfrac{(-72+54i)\div 2}{100\div 2}\Rightarrow \\ \)
    \(\dfrac{-36+27i}{50}\)
41. \(\dfrac{5i}{-6-i}\cdot \dfrac{-6+i}{-6+i}\Rightarrow \dfrac{-30i+5i^2}{36-i^2}\Rightarrow \dfrac{-30i-5}{36+1}\Rightarrow \dfrac{-30i-5}{37}\)
42. \(\dfrac{8i}{6-7i}\cdot \dfrac{6+7i}{6+7i}\Rightarrow \dfrac{48i+56i^2}{36-49i^2}\Rightarrow \dfrac{48i-56}{36+49}\Rightarrow \dfrac{48i-56}{85}\)
43. \(\pm 9i\)
44. \(\sqrt{-5\cdot 9}\Rightarrow \pm 3\sqrt{5}i\)
45. \(\sqrt{-20}\Rightarrow \sqrt{-4\cdot 5}\Rightarrow \pm2\sqrt{5}i\)
46. \(\sqrt{24}\Rightarrow \sqrt{4\cdot 6}\Rightarrow \pm 2\sqrt{6}\)
47. \(\dfrac{3+\sqrt{3\cdot -9}}{6}\Rightarrow \dfrac{3+3\sqrt{3}i}{6}\Rightarrow \dfrac{1+\sqrt{3}i}{2}\)
48. \(\dfrac{-4-\sqrt{-2\cdot 4}}{-4}\Rightarrow \dfrac{-4-2\sqrt{2}i}{-4}\Rightarrow \dfrac{2+\sqrt{2}i}{2}\)
49. \(\dfrac{8-4i}{4}\Rightarrow 2-i\)
50. \(\dfrac{6+\sqrt{2\cdot -16}}{4}\Rightarrow \dfrac{6+4\sqrt{2}i}{4}\Rightarrow \dfrac{3+2\sqrt{2}i}{2}\)
51. \(i\)
52. \(i^3\Rightarrow -i\)
53. 1
54. 1
55. \(i^2\Rightarrow -1\)
56. \(i\)
57. \(i^2\Rightarrow -1\)
58. \(i^3\Rightarrow -i\)