Solutions to Odd-Numbered Answers

1. a is the real part, b is the imaginary part, and $i=\sqrt{−1}$

3. Polar form converts the real and imaginary part of the complex number in polar form using $x=r\cos\theta$ and $y=r\sin\theta$

5. $z^{n}=r^{n}\left(\cos\left(n\theta\right)+i\sin\left(n\theta\right)\right)$. It is used to simplify polar form when a number has been raised to a power.

7. $5\sqrt{2}$

9. $\sqrt{38}$

11. $\sqrt{14.45}$

13. $4\sqrt{5}\text{cis}\left(333.4^{\circ}\right)$

15. $2\text{cis}\left(\frac{\pi}{6}\right)$

17. $\frac{7\sqrt{3}}{2}+i\frac{7}{2}$

19. $−2\sqrt{3}−2i$

21. $−1.5−i\frac{3\sqrt{3}}{2}$

23. $4\sqrt{3}\text{cis}\left(198^{\circ}\right)$

25. $\frac{3}{4}\text{cis}\left(180^{\circ}\right)$

27. $5\sqrt{3}\text{cis}\left(\frac{17\pi}{24}\right)$

29. $7\text{cis}\left(70^{\circ}\right)$

31. $5\text{cis}\left(80^{\circ}\right)$

33. $5\text{cis}\left(\frac{\pi}{3}\right)$

35. $125\text{cis}\left(135^{\circ}\right)$

37. $9\text{cis}\left(240^{\circ}\right)$

39. $\text{cis}\left(\frac{3\pi}{4}\right)$

41. $3\text{cis}\left(80^{\circ}\right)\text{, }3\text{cis}\left(200^{\circ}\right)\text{, }3\text{cis}\left(320^{\circ}\right)$

43. $2\sqrt[3]{4}\text{cis}\left(\frac{2\pi}{9}\right)\text{, }2\sqrt[3]{4}\text{cis}\left(\frac{8\pi}{9}\right)\text{, }2\sqrt[3]{4}\text{cis}\left(\frac{14\pi}{9}\right)$

45. $2\sqrt{2}\text{cis}\left(\frac{7\pi}{8}\right)\text{, }2\sqrt{2}\text{cis}\left(\frac{15\pi}{8}\right)$

47.

49.

51.

53.

55.

57. $3.61e^{−0.59i}$

59. $−2+3.46i$

61. $−4.33−2.50i$