Answer
$ \sqrt[4] 8(cos75^\circ+i\ sin75^\circ)$, $ \sqrt[4] 8(cos165^\circ+i\ sin165^\circ)$, $ \sqrt[4] 8(cos255^\circ+i\ sin255^\circ)$, $ \sqrt[4] 8(cos345^\circ+i\ sin345^\circ)$.
Work Step by Step
Based on the given conditions, we have:
$4-4\sqrt 3i=8(cos300^\circ+i\ sin300^\circ)$, $(4-4\sqrt 3i)^{1/4}=\sqrt[4] 8(cos(\frac{360k+300}{4})^\circ+i\ sin(\frac{360k+300}{4})^\circ)$, $k=0, z_0=\sqrt[4] 8(cos75^\circ+i\ sin75^\circ)$, $k=1, z_1=\sqrt[4] 8(cos165^\circ+i\ sin165^\circ)$, $k=2, z_2=\sqrt[4] 8(cos255^\circ+i\ sin255^\circ)$, $k=3, z_3=\sqrt[4] 8(cos345^\circ+i\ sin345^\circ)$.
