Help me solve these three questions.

Question

Polar Coordinates and Polar Form of Complex numbers

Show your calculations in detail.

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°)

Kindly help me with the calculation step by step.

Answer 1

#1 Using DeMoivre's Theorem, one of the cube roots is 27^1/3cis(240°/3) = 3cis80°

The 3 cube roots of the given number are equally spaced around the circle centered at (0,0) with radius 3.

360° / 3 = 120°

Another cube root is 3cis(80°+120°) = 3cis200°

The third cube root is 3cis(80°+2(120°) = 3cis320°

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#2 convert to polar form to get 5√3cis30°

So, [5√3cis30°]¹⁰ = (5√3)¹⁰cis[10(30°)] = 237046875cis300°

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#3 Divide 21 by 3 and subtract 75° from 135°

So, z₁ / z₂ = 7cis60°