10.5E: Polar Form of Complex Numbers (Exercises)

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For the following exercises, find the absolute value of each complex number.

28. \(-2+6 \mathbf{i}\)

29. \(4-3 \mathbf{i}\)

Write the complex number in polar form.

30. \(5+9 \mathbf{i}\)

31. \(\frac{1}{2}-\frac{\sqrt{3}}{2} \mathbf{i}\)

For the following exercises, convert the complex number from polar to rectangular form.

32. \(z=5 \operatorname{cis}\left(\frac{5 \pi}{6}\right)\)

33. \(z=3 \operatorname{cis}\left(40^{\circ}\right)\)

For the following exercises, find the product \(z_{1} z_{2}\) in polar form.

34.

\(z_{1}=2 \operatorname{cis}\left(89^{\circ}\right)\)

\(z_{2}=5 \operatorname{cis}\left(23^{\circ}\right)\)

35.

\(z_{1}=10\) cis \(\left(\frac{\pi}{6}\right)\)

\(z_{2}=6 \operatorname{cis}\left(\frac{\pi}{3}\right)\)

For the following exercises, find the quotient \(\frac{z_{1}}{z_{2}}\) in polar form.

36. \(z_{1}=12 \operatorname{cis}\left(55^{\circ}\right)\)

\(z_{2}=3 \operatorname{cis}\left(18^{\circ}\right)\)

37. \(z_{1}=27 \operatorname{cis}\left(\frac{5 \pi}{3}\right)\)

\(z_{2}=9 \operatorname{cis}\left(\frac{\pi}{3}\right)\)

For the following exercises, find the powers of each complex number in polar form.

38. Find \(z^{4}\) when \(z=2\) cis \(\left(70^{\circ}\right)\)

39. Find \(z^{2}\) when \(z=5\) cis \(\left(\frac{3 \pi}{4}\right)\)

For the following exercises, evaluate each root.

40. Evaluate the cube root of \(z\) when \(z=64\) cis \(\left(210^{\circ}\right)\).

41. Evaluate the square root of \(z\) when \(z=25\) cis \(\left(\frac{3 \pi}{2}\right)\).

For the following exercises, plot the complex number in the complex plane.

42. \(6-2 \mathbf{i}\)

43. \(-1+3 \mathbf{i}\)