Roots of Complex Number/Examples/4th Roots of -16 i

Example of Roots of Complex Number: Corollary
The complex $4$th roots of $-16 i$ are given by:
 * $\paren {-16 i}^{1/4} = \set {2 \, \map \cis {67.5 + 90 k} \degrees}$

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

That is:

Proof

 * Complex 4th Roots of -16 i.png

Let $z^4 = -16 i$.

We have that:
 * $z^4 = 16 \, \map \cis {\dfrac {3 \pi} 2 + 2 k \pi}$

Let $z = r \cis \theta$.

Then: