Consider the calculation of roots of an equation z N = where N 1 is

Consider the calculation of roots of an equation z^N = α where N ≥ 1 is an integer and α = |α|e^jϕ a nonzero complex numb

(a) First verify that there are exactly N roots for this equation and that they are given by

z_k = re^jθk where r = |α|^1/N and θ_k = (ϕ + 2πk)/N for k = 0, 1, ... , N – 1.

(b) Use the above result to find the roots of the following equations

(i) z² = 1; (ii) z² = -1; (iii) z³ = 1; (iv) z³ = -1.

and plot them in a polar plane (i..e., indicating their magnitude and phase). Explain how the roots are distributed in the polar plane.