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If-2-1-10-cos-9-i-sin-9-Find-z-5-




Question Number 206695 by hardmath last updated on 22/Apr/24
If   (2)^(1/(10))  (cos 9° + i sin 9°)  Find:   z^5  = ?
$$\mathrm{If}\:\:\:\sqrt[{\mathrm{10}}]{\mathrm{2}}\:\left(\mathrm{cos}\:\mathrm{9}°\:+\:\boldsymbol{\mathrm{i}}\:\mathrm{sin}\:\mathrm{9}°\right) \\ $$$$\mathrm{Find}:\:\:\:\boldsymbol{\mathrm{z}}^{\mathrm{5}} \:=\:? \\ $$
Answered by A5T last updated on 22/Apr/24
z=(2)^(1/(10)) e^(i(9°)) ⇒z^5 =(√2)e^(i(45°)) =(√2)(cos45+isin45)  =1+i
$${z}=\sqrt[{\mathrm{10}}]{\mathrm{2}}{e}^{{i}\left(\mathrm{9}°\right)} \Rightarrow{z}^{\mathrm{5}} =\sqrt{\mathrm{2}}{e}^{{i}\left(\mathrm{45}°\right)} =\sqrt{\mathrm{2}}\left({cos}\mathrm{45}+{isin}\mathrm{45}\right) \\ $$$$=\mathrm{1}+{i} \\ $$

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