Roots of Complex Number/Examples/z^4 + 81 = 0

Theorem
The roots of the polynomial:
 * $z^4 + 81$

are:
 * $\set {3 \cis 45 \degrees, 3 \cis 135 \degrees, 3 \cis 225 \degrees, 3 \cis 315 \degrees}$

Proof
From Factorisation of $z^n + 1$:


 * $z^4 + 1 = \displaystyle \prod_{k \mathop = 0}^3 \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} 4}$

Thus:
 * $z = \set {\exp \dfrac {\paren {2 k + 1} i \pi} 4}$