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Question Number 190732 by zinaung last updated on 10/Apr/23

Answered by JDamian last updated on 10/Apr/23

$$is\sqrt{-\sqrt{3+i}}or\sqrt{-\sqrt{3}+i}?$$

Answered by Frix last updated on 10/Apr/23

$$\sqrt{-\sqrt{3}+\mathrm{i}}=\sqrt{{2\mathrm{e}}^{\mathrm{i}\frac{5\pi }{6}}}=\sqrt{2}{\mathrm{e}}^{\mathrm{i}\frac{5\pi }{12}}=\frac{\sqrt{3}-1}{2}+\frac{\sqrt{3}+1}{2}\mathrm{i}$$

Answered by Frix last updated on 14/Apr/23

$$\sqrt{-\sqrt{3+\mathrm{i}}}=\sqrt{-\sqrt{\sqrt{10}{\mathrm{e}}^{\mathrm{i}{\tan }^{-1}\frac{1}{3}}}}=$$$$=\sqrt{-\sqrt[4]{10}{\mathrm{e}}^{\mathrm{i}\frac{{\tan }^{-1}\frac{1}{3}}{2}}}=$$$$=\sqrt{\sqrt[4]{10}{\mathrm{e}}^{\mathrm{i}\frac{2\pi +{\tan }^{-1}\frac{1}{3}}{2}}}=$$$$=\sqrt[8]{10}{\mathrm{e}}^{\mathrm{i}\frac{2\pi +{\tan }^{-1}\frac{1}{3}}{4}}$$

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