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Question Number 212506 by hardmath last updated on 15/Oct/24

$$\mathrm{Find}:\sqrt[4]{-\frac{18}{1+\mathbf{i}\sqrt{3}}}$$

Answered by Frix last updated on 15/Oct/24

$$-\frac{18}{1+\sqrt{3}\mathrm{i}}=-\frac{9}{2}+\frac{9\sqrt{3}}{2}\mathrm{i}={9\mathrm{e}}^{\mathrm{i}\frac{2\pi }{3}}$$$${\left({9\mathrm{e}}^{\mathrm{i}\frac{2\pi }{3}}\right)}^{\frac{1}{4}}=\sqrt{3}{\mathrm{e}}^{\mathrm{i}\frac{\pi }{6}}=\frac{3}{2}+\frac{\sqrt{3}}{2}\mathrm{i}$$

Commented by hardmath last updated on 15/Oct/24

$$\mathrm{thankyou}\mathrm{dearprofessor}$$

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