Question

More important than the math, I would really like to see the graph of the 5 point in the complex plane. Thank you!

#2)(20pts) In polar form write the 5 distinct roots of 3 Draw the 5 points in the complex plane

Answer 1

z^5 = 2^5 (- 1/2 + i squareroot 3/2) First of all we will try to change it in to the polar from i.e z = re^itheta where z = x + iy theta = tan^-1 (y/x) and r = squareroot x^2 + y^2 here r = 2 z^5 = 2^5 e^i(5 pi /12 + 2k pi) where k = 0, 1, 2, 3, 4 Implies z = [2^5 e^i(5 pi /12 + 2k pi)]^1/5 Implies z = 2e^i(5 pi /12 + 2k pi)times 1/5, k = 0, 1, 2, 3, 4 Implies z = 2e^i(pi/12 + 2k pi /5) k = 0, 1, 2, 3, 4 \r\n So five Roots are as following: all the five Roots are on the circle of radius r = 2 are given by z_1, z_2, z_3, z_4, z_5 where (i) z_1 is at k = 0 So z_1 = 2e^i(pi /12) (ii) and z_2 is at k = 1 z_2 = 2e^i(pi /12 + 2 pi /5) Implies z_2 = 2e^i(22/60 pi) (iii) z_3 at k