Roots of Complex Number/Examples/Cube Roots of -11-2i

Example of Roots of Complex Number: Corollary
The complex cube roots of $-11 - 2 i$ are given by:
 * $\paren {-11 - 2 i}^{1/3} = \set {1 + 2 i, \dfrac 1 2 - \sqrt 3 + \paren {1 + \dfrac {\sqrt 3} 2} i, -\dfrac 1 2 - \sqrt 3 + \paren {\dfrac {\sqrt 3} 2 - 1} i}$

Proof
Let $z^3 = -11 - 2 i$.

We have that:
 * $\cmod {z^3} = \sqrt {125}$

so we look for a $z$ such that $\cmod z = \sqrt 5$ and such that $\map \arg z$ is somewhere in the first quadrant.

From the geometry of the situation, the obvious first such possible number to investigate is $z = 1 + 2 i$.

We have:

So $z = 1 + 2 i$ is a complex cube root of $-11 - 2 i$.