Question

29. Roots and Factors. For each of the following find the roots of the given equations and sketch the roots in the complex plane: (a) cube roots z3 = 1 (b) square roots z2 = i (c) sixth roots z6 = -64 (d) fifth roots z5 = 32e5Ti/3

Answer 1

(a) Z^3 = 1 = cos 2 k h^- + i sin 2 k h^- , k epsilon Z = e^1 2k h^- = e^i 2 k h^-/3 = cos 2 k h^-/2 + i sin 2 k h^-/3 if k = 0, mod 3, then, k = 3m, say = cos 2 m h^- + i sin 2 m h^- = 1. if k = 1 mod 3, then, k = 3m + 1, say Therefore = cos (2 m h^- + 2 h^-/3) + i sin (2 m h^- + 2 h^-/3) = -1/2 + i squareroot 3/2 if k = 2 mod 3, then k = 3 m + 2, say Therefore = cos (2 m h^- + 3 h^-/3) + i sin (2 m h^- + 4 h^-/3) = -1/2 - i squareroot - i squareroot 3/2. Therefore z = 1, -1/2 + i squareroot 3/2 -1/2 - i squareroot 3/2 b) z^2 = i = cos (4k + 1) h^-/2 + i sin (4k + 1) h^-/2, k epsilon Z