How do I solve these problems.

Question

Polar Coordinates and Polar Form of Complex numbers

Help me solve the following questions, I have no idea.

1. Evaluate the cube root of z = 27cis(240°). Then raise them to the cube. Show the steps of your reasoning.

2. Evaluate [ 5√3 (√3/2 + i/2) ]¹⁰

3. Find z1 / z2 in polar form:

Z1 = 21cis(135°) and Z2 = 3cis(75°)

Answer 1

z=rcisθ (rcisθ)^1/n=r^1/n(cis((θ+(360k)/n)) k=0, 1, 2, ... n-1

First find (θ/n) and (360k)/n 240/3=80 and 360(0)=0, 360(1)/3 = 120 and 720/3 = 240

So to find (27cis240)^1/3, take 27^1/3=3 and find cos80 and sin80, cos(80+120) and sin(80+120), cos(80+240) and sin(80+240).

z^1/3=3cis(80), 3cis(200), and 3cis(320).

(Notice that the angles 80°, 200°, and 320° are all 120° apart)

z³=r³cis(3θ) so (3cis(80))³=3³cis(240)

Hope this helped.