Let and be two roots of x^2+2x+2=0.

Question

Let $\alpha$ and $\beta$ be two roots of $x^2+2x+2=0$. Then $\alpha^{15}+\beta^{15}$ is equal to:

Accepted Answer

$\alpha,\beta=-1\pm i=\sqrt2e^{\pm3i\pi/4}$.

$$\alpha^{15}+\beta^{15}=(\sqrt2)^{15}\left(e^{i45\pi/4}+e^{-i45\pi/4}\right)=(\sqrt2)^{15}\cdot2\cos\frac{3\pi}4=2(\sqrt2)^{14}\cdot\left(-\frac1{\sqrt2}\right)=-256.$$

Solutions · 1