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Question Number 91452 by  M±th+et+s last updated on 30/Apr/20

$$\sqrt[4]{-1}=?$$

Commented by  M±th+et+s last updated on 30/Apr/20

$$thanxforsolutions$$

Answered by mr W last updated on 30/Apr/20

$$\sqrt[4]{-1}=z$$$$z^4=-1=\cos \pi +i\sin \pi$$$$z=\cos (\frac{k\pi }{2}+\frac{\pi }{4})+i\sin (\frac{k\pi }{2}+\frac{\pi }{4})(k=0,1,2,3)$$$$=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$$$$or$$$$=-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$$$$or$$$$=+\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i$$$$or$$$$=-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i$$

Answered by MJS last updated on 30/Apr/20

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

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