find all the cube roots of 4surd 2(i-1)

Question

4√2(¡-1)

Answer 1

z = 2^5/2(-1 + i) = 2^5/2e^i(¾)(pi)   

or 2^5/2 [cos θ + i sin θ]  where θ = (¾)(pi) + 2(pi)k, k = integer.

Therefore the three cube roots are 2^5/6[cos φ + i sin φ] for φ = = 45⁰ or 165^o or 285^o

corresponding to k= 1, 2 or 2 and converting from radians to degrees, and using DeMoivre's theorem.