Answer
$ \sqrt[6] 2(cos15^\circ+i\ sin15^\circ)$, $ \sqrt[6] 2(cos135^\circ+i\ sin135^\circ)$, $ \sqrt[6] 2(cos255^\circ+i\ sin255^\circ)$.
Work Step by Step
Based on the given conditions, we have:
$1+i=\sqrt 2(cos45^\circ+i\ sin45^\circ)$, $(1+i)^{1/3}=\sqrt[6] 2(cos(\frac{360k+45}{3})^\circ+i\ sin(\frac{360k+45}{3})^\circ)$, $k=0, z_0=\sqrt[6] 2(cos15^\circ+i\ sin15^\circ)$, $k=1, z_1=\sqrt[6] 2(cos135^\circ+i\ sin135^\circ)$, $k=2, z_2=\sqrt[6] 2(cos255^\circ+i\ sin255^\circ)$.
