Question Number 206695 by hardmath last updated on 22/Apr/24
$$\mathrm{If}\:\:\:\sqrt[{\mathrm{10}}]{\mathrm{2}}\:\left(\mathrm{cos}\:\mathrm{9}°\:+\:\boldsymbol{\mathrm{i}}\:\mathrm{sin}\:\mathrm{9}°\right) \\ $$$$\mathrm{Find}:\:\:\:\boldsymbol{\mathrm{z}}^{\mathrm{5}} \:=\:? \\ $$
Answered by A5T last updated on 22/Apr/24
$${z}=\sqrt[{\mathrm{10}}]{\mathrm{2}}{e}^{{i}\left(\mathrm{9}°\right)} \Rightarrow{z}^{\mathrm{5}} =\sqrt{\mathrm{2}}{e}^{{i}\left(\mathrm{45}°\right)} =\sqrt{\mathrm{2}}\left({cos}\mathrm{45}+{isin}\mathrm{45}\right) \\ $$$$=\mathrm{1}+{i} \\ $$