De Moivre’s theorem

Question

Use De Moivre’s theorem and determine the value of...

[8(-1 + 3^1/2 i)]^1/4

Do the calculation in polar form but give the answer in rectangular form.

Answer 1

|| [8( - 1 + 8i√3 || = 16

θ = 120º

z^1/4 = 16^1/4[cos [(120º + 360k) / 4] } i sin [(120º + 360k) / 4)], k = 0, 1, 2, and 3.

Since 16^1/4 = ±2, 8 roots exist.