Roots of Complex Number/Corollary/Examples/Fourth Roots of 2-2i

Example of Roots of Complex Number: Corollary
The complex $4$th roots of $2 - 2 i$ are given by:
 * $\paren {2 - 2 i}^{1/4} = \set {b, bi, -b, -bi}$

where:
 * $b = \sqrt [8] 8 \paren {\cos \dfrac \pi {16} + i \sin \dfrac \pi {16} }$

Proof
Let $z = 2 - 2 i$.

Then:

and so:
 * $\arg z = -\dfrac \pi 4$

Let $b$ be defined as:

Then we have that the complex $4$th roots of unity are:
 * $1, i, -1, -i$

The result follows from Roots of Complex Number: Corollary.