Roots of Complex Number/Examples/4th Roots of -2 root 3 - 2i

Example of Roots of Complex Number: Corollary
The complex $4$th roots of $-2 \sqrt 3 - 2 i$ are given by:
 * $\paren {-2 \sqrt 3 - 2 i}^{1/4} = \set {\sqrt 2 \paren {\map \cis {\dfrac {7 \pi} {24} + \dfrac {k \pi} 2} } }$

for $k = 0, 1, 2, 3$.

That is:

Proof

 * Complex 4th Roots of -2 root 3 - 2i.png

Let $z^4 = -2 \sqrt 3 - 2 i$.

We have that:
 * $z^4 = 4 \paren {\map \cis {\dfrac {7 \pi} 6 + 2 k \pi} }$

Let $z = r \cis \theta$.

Then: