Answer
$\pm1$, $\frac{1}{2}+\frac{\sqrt 3}{2}i$, $-\frac{1}{2}+\frac{\sqrt 3}{2}i$, $ -\frac{1}{2}-\frac{\sqrt 3}{2}i$, $ \frac{1}{2}-\frac{\sqrt 3}{2}i$. See graph.
Work Step by Step
Based on the given conditions, we have:
$1=cos0^\circ+i\ sin0^\circ)$, $(i)^{1/6}=cos(\frac{360k+0}{6})^\circ+i\ sin(\frac{360k+0}{6})^\circ)$, $k=0, z_0= cos0^\circ+i\ sin0^\circ=1$, $k=1, z_1= cos60^\circ+i\ sin60^\circ=\frac{1}{2}+\frac{\sqrt 3}{2}i$, $k=2, z_2= cos120^\circ+i\ sin120^\circ=-\frac{1}{2}+\frac{\sqrt 3}{2}i$, $k=3, z_3= cos180^\circ+i\ sin180^\circ=-1$, $k=4, z_4= cos240^\circ+i\ sin240^\circ=-\frac{1}{2}-\frac{\sqrt 3}{2}i$, $k=5, z_5= cos300^\circ+i\ sin300^\circ=\frac{1}{2}-\frac{\sqrt 3}{2}i$.
See graph.
