Roots of Complex Number/Examples/Cube Roots of 2+2i

Example of Roots of Complex Number: Corollary
The complex cube roots of $2 + 2 i$ are given by:
 * $\paren {2 + 2 i}^{1/3} = \set {\sqrt 2 \paren {\cos \dfrac \pi {12} + i \sin \dfrac \pi {12} }, -1 + i, -\sqrt 2 \paren {\cos \dfrac {5 \pi} {12} + i \sin \dfrac {5 \pi} {12} }}$

Proof
Let $z = 2 + 2 i$.

Then: Then:

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

Let $b$ be defined as:

Then we have that the complex cube roots of unity are:
 * $1, \exp {\dfrac {2 i \pi} 3}, \exp {\dfrac {-2 i \pi} 3}$

Thus from Roots of Complex Number: Corollary: