Answer to Question #117390 in Complex Analysis for Amoah Henry

Question

Express −1+i in polar form. Hence show that (−1+i)16 is real and that 1/(−1+i)6 is purely imaginary, giving the value for each.

Answer 1

$-1+i=\sqrt{2}(\cos \frac{3\pi}{4}+i\sin \frac{3\pi}{4})$

$(-1+i)^{16}=256(\cos \frac{48\pi}{4}+i\sin \frac{48\pi}{4})=256(\cos 12\pi+i\sin 12\pi)=256.$

$\frac{1}{(-1+i)^6}=(-1+i)^{-6}=\frac{1}{8}(\cos \frac{18\pi}{4}+i\sin \frac{18\pi}{4})=\frac{1}{8}(\cos \frac{9\pi}{2}+i\sin \frac{9\pi}{2})=\frac{1}{8}i$.