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Question Number 155899 by Tawa11 last updated on 05/Oct/21

$$\mathrm{Find}\mathrm{the}\mathrm{nth}\mathrm{root}\mathrm{of}1.\mathrm{Find}\mathrm{these}\mathrm{values}\mathrm{when}\mathrm{n}=6.$$

Answered by Ar Brandon last updated on 05/Oct/21

$$1=e^{2k\pi i},k\in \mathbb{Z}$$$$z_n=\sqrt[n]{1}=e^{\frac{2k\pi }{n}i},k\in [0,n-1]$$$$z_6=e^{\frac{k\pi }{3}i},k\in [0,5]$$$$z_6=1,\frac{1}{2}(1+i\sqrt{3}),-\frac{1}{2}(1-i\sqrt{3})$$$$-1,-\frac{1}{2}(1+i\sqrt{3}),\frac{1}{2}(1-i\sqrt{3})$$

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