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Answer Key 9.9

  1. [latex]3+8-4i[/latex]
    [latex]11-4i[/latex]
  2. [latex]3i-7i[/latex]
    [latex]-4i[/latex]
  3. [latex]7i-3+2i[/latex]
    [latex]9i-3[/latex]
  4. [latex]5-6-6i[/latex]
    [latex]-1-6i[/latex]
  5. [latex]-6i-3-7i[/latex]
    [latex]-13i-3[/latex]
  6. [latex]-8i-7i-5+3i[/latex]
    [latex]-12i-5[/latex]
  7. [latex]3-3i-7-8i[/latex]
    [latex]-4-11i[/latex]
  8. [latex]-4-i+1-5i[/latex]
    [latex]-3-6i[/latex]
  9. [latex]i-2-3i-6[/latex]
    [latex]-2i-8[/latex]
  10. [latex]5-4i+8-4i[/latex]
    [latex]13-8i[/latex]
  11. [latex]-48i^2[/latex]
    [latex]-48(-1)[/latex]
    [latex]48[/latex]
  12. [latex]-24i^2[/latex]
    [latex]-24(-1)[/latex]
    [latex]24[/latex]
  13. [latex]-40i^2[/latex]
    [latex]-40(-1)[/latex]
    [latex]40[/latex]
  14. [latex]-32i^2[/latex]
    [latex]-32(-1)[/latex]
    [latex]32[/latex]
  15. [latex]49i^2[/latex]
    [latex]49(-1)[/latex]
    [latex]-49[/latex]
  16. [latex]-7i^2(4-3i)[/latex]
    [latex]-28i^2+21i^3[/latex]
    [latex]-28(-1)+21(-1)i[/latex]
    [latex]28-21i[/latex]
  17. [latex]36+60i+25i^2[/latex]
    [latex]36+60i+25(-1)[/latex]
    [latex]36+60i-25[/latex]
    [latex]11+60i[/latex]
  18. [latex]16i^2(-2-8i)[/latex]
    [latex]32i^2+128i^3[/latex]
    [latex]32(-1)+128(-1)i[/latex]
    [latex]-32-128i[/latex]
  1. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrl} &56&-&42i&& \\ &&+&32i&-&24i^2 \\ \hline &56&-&10i&-&24(-1) \\ +&24&&&\Longleftarrow & \\ \hline &80&-&10i&& \end{array}[/latex]
  2. [latex]9i^2(4-4i)[/latex]
    [latex]-36i^2+36i^3[/latex]
    [latex]-36(-1)+36(-1)i[/latex]
    [latex]36-36i[/latex]
  3. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrl} &-8&+&10i&& \\ &&+&28i&-&35i^2 \\ \hline &-8&+&38i&-&35(-1) \\ +&35&&&\Longleftarrow & \\ \hline &27&+&38i&& \end{array}[/latex]
  4. [latex]-32i+64i+4+12i[/latex]
    [latex]-28+76i[/latex]
  5. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrl} &32&+&24i&& \\ &&-&16i&-&12i^2 \\ \hline &32&+&8i&-&12(-1) \\ +&12&&&\Longleftarrow & \\ \hline &44&+&8i&& \end{array}[/latex]
  6. [latex]-18i+12i^2-28i^2[/latex]
    [latex]-18i+12(-1)-28(-1)[/latex]
    [latex]-18i-12+28[/latex]
    [latex]-18i+16[/latex]
  7. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrl} &2&+&10i&& \\ &&+&i&+&5i^2 \\ \hline &2&+&11i&+&5(-1) \\ -&5&&&\Longleftarrow & \\ \hline &-3&+&11i&& \end{array}[/latex]
  8. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrl} &-6&+&3i&& \\ &&+&10i&-&5i^2 \\ \hline &-6&+&13i&-&5(-1) \\ +&5&&&\Longleftarrow & \\ \hline &-1&+&13i&& \end{array}[/latex]
  9. [latex]\dfrac{-9+5i}{i}\cdot \dfrac{i}{i}\Rightarrow \dfrac{-9i+5i^2}{i^2}\Rightarrow \dfrac{-9i+5(-1)}{(-1)}\Rightarrow 9i+5[/latex]
  10. [latex]\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}[/latex]
  11. [latex]\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}[/latex]
  12. [latex]\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}[/latex]
  13. [latex]\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}[/latex]
  14. [latex]\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}[/latex]
  15. [latex]\dfrac{10-i}{-i}\cdot \dfrac{i}{i}\Rightarrow \dfrac{10i-i^2}{-i^2}\Rightarrow \dfrac{10i-(-1)}{-(-1)}\Rightarrow 10i+1[/latex]
  16. [latex]\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[/latex]
  17. [latex]\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}[/latex]
  18. [latex]\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[/latex]
    [latex]\dfrac{9i-45}{26}[/latex]
  19. [latex]\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}[/latex]
  20. [latex]\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}[/latex]
  21. [latex]\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}[/latex]
  22. [latex]\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[/latex]
    [latex]\dfrac{-36+27i}{50}[/latex]
  23. [latex]\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}[/latex]
  24. [latex]\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}[/latex]
  25. [latex]\pm 9i[/latex]
  26. [latex]\sqrt{-5\cdot 9}\Rightarrow \pm 3\sqrt{5}i[/latex]
  27. [latex]\sqrt{-20}\Rightarrow \sqrt{-4\cdot 5}\Rightarrow \pm2\sqrt{5}i[/latex]
  28. [latex]\sqrt{24}\Rightarrow \sqrt{4\cdot 6}\Rightarrow \pm 2\sqrt{6}[/latex]
  29. [latex]\dfrac{3+\sqrt{3\cdot -9}}{6}\Rightarrow \dfrac{3+3\sqrt{3}i}{6}\Rightarrow \dfrac{1+\sqrt{3}i}{2}[/latex]
  30. [latex]\dfrac{-4-\sqrt{-2\cdot 4}}{-4}\Rightarrow \dfrac{-4-2\sqrt{2}i}{-4}\Rightarrow \dfrac{2+\sqrt{2}i}{2}[/latex]
  31. [latex]\dfrac{8-4i}{4}\Rightarrow 2-i[/latex]
  32. [latex]\dfrac{6+\sqrt{2\cdot -16}}{4}\Rightarrow \dfrac{6+4\sqrt{2}i}{4}\Rightarrow \dfrac{3+2\sqrt{2}i}{2}[/latex]
  1. [latex]i[/latex]
  2. [latex]i^3\Rightarrow -i[/latex]
  3. 1
  4. 1
  5. [latex]i^2\Rightarrow -1[/latex]
  6. [latex]i[/latex]
  7. [latex]i^2\Rightarrow -1[/latex]
  8. [latex]i^3\Rightarrow -i[/latex]

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