Roots of Complex Number/Examples/Square Roots of 4 root 2 + 4 root 2 i

Example of Roots of Complex Number
The complex square roots of $4 \sqrt 2 + 4 \sqrt 2 i$ are given by:
 * $\paren {4 \sqrt 2 + 4 \sqrt 2 i}^{1/2} = \set {\sqrt 8 \cis 22.5 \degrees, \sqrt 8 \cis 202.5 \degrees}$

Proof

 * Complex Square Roots of 4 root 2 + 4 root 2 i.png

Let $z^2 = 4 \sqrt 2 + 4 \sqrt 2 i$.

We have that:
 * $z^2 = 8 \paren {\dfrac {\sqrt 2} 2 + \dfrac {\sqrt 2} 2 i}$

and it is seen that:
 * $\dfrac {\sqrt 2} 2 + \dfrac {\sqrt 2} 2 i = \cis \dfrac \pi 4$

Hence