Roots of Complex Number/Examples/5th Roots of -4 + 4i

Example of Roots of Complex Number
The complex $5$th roots of $-4 + 4i$ are given by:
 * $\paren {-4 + 4i}^{1/5} = \set {\sqrt 2 \, \map \cis {27 + 72 k} \degrees}$

for $k = 0, 1, 2, 3, 4$.

That is:

Proof

 * Complex 5th Roots of -4 + 4i.png

Let $z^5 = -4 + 4 i$.

We have that:
 * $z^5 = 4 \sqrt 2 \, \map \cis {\dfrac {3 \pi} 4 + 2 k \pi} = 4 \sqrt 2 \, \map \cis {135 \degrees + k \times 360 \degrees}$

Let $z = r \cis \theta$.

Then: