Roots of Complex Number/Examples/Square Roots of 2 root 3 - 2i

Example of Roots of Complex Number
The complex square roots of $2 \sqrt 3 - 2 i$ are given by:
 * $\paren {2 \sqrt 3 - 2 i}^{1/2} = \set {2 \cis 165 \degrees, 2 \cis 345 \degrees}$

Proof

 * Complex Square Roots of 2 root 3 - 2i.png

Let $z^2 = 2 \sqrt 3 - 2 i$.

We have that:
 * $z^2 = 4 \paren {\dfrac {\sqrt 3} 2 - \dfrac 1 2 i}$

and it is seen (indirectly) from Cube Roots of Unity that:
 * $\dfrac {\sqrt 3} 2 - \dfrac 1 2 i = \cis \dfrac {11 \pi} 6$

Hence