Question Number 212506 by hardmath last updated on 15/Oct/24
$$\mathrm{Find}:\:\:\:\:\:\:\sqrt[{\mathrm{4}}]{−\:\frac{\mathrm{18}}{\mathrm{1}\:+\:\boldsymbol{\mathrm{i}}\:\sqrt{\mathrm{3}}}} \\ $$
Answered by Frix last updated on 15/Oct/24
$$−\frac{\mathrm{18}}{\mathrm{1}+\sqrt{\mathrm{3}}\mathrm{i}}=−\frac{\mathrm{9}}{\mathrm{2}}+\frac{\mathrm{9}\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{i}=\mathrm{9e}^{\mathrm{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \\ $$$$\left(\mathrm{9e}^{\mathrm{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} =\sqrt{\mathrm{3}}\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{6}}} =\frac{\mathrm{3}}{\mathrm{2}}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{i} \\ $$
Commented by hardmath last updated on 15/Oct/24
$$\mathrm{thankyou}\:\mathrm{dearprofessor} \\ $$