Question

Find the values of $(i{)}^{\frac{1}{3}}$

Answer 1

$(i{)}^{\frac{1}{3}}={(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})}^{\frac{1}{3}}$ ....$[\because i=\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}]$
$={[\cos (2k\pi +\frac{\pi }{2})+i\sin (2k\pi +\frac{\pi }{2})]}^{\frac{1}{3}}$
$=\cos \frac{2k\pi +\frac{\pi }{2}}{3}+i\sin \frac{2k\pi +\frac{\pi }{2}}{3}$, where $k=0,1,2$
$=\cos \frac{4k\pi +\pi }{6}+i\sin \frac{4k\pi +\pi }{6}$ Where $k=0,1,2$
$=\cos \frac{\pi }{6}(4k+1)+i\sin \frac{\pi }{6}(4k+1)$ Wheree $k=0,1,2$
The values are $\cos \frac{\pi }{6}+i\sin \frac{\pi }{6},\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6},\cos \frac{9\pi }{6}+i\sin \frac{9\pi }{6}$.