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Question Number 19312 by Tinkutara last updated on 09/Aug/17

$$\mathrm{Product}\mathrm{of}n,n^{\mathrm{th}}\mathrm{roots}\mathrm{of}\mathrm{unity}$$$$=1.\alpha .{\alpha }^2.{\alpha }^3.....{\alpha }^{n-1}={(-1)}^{n-1}$$$$\mathrm{Why}?\mathrm{How}\mathrm{to}\mathrm{get}\mathrm{RHS}?$$

Answered by ajfour last updated on 09/Aug/17

$${\mathrm{x}}^{\mathrm{n}}-1=(\mathrm{x}-1)(\mathrm{x}-\alpha )(\mathrm{x}-{\alpha }^2)...(\mathrm{x}-{\alpha }^{\mathrm{n}-1})$$$$\mathrm{comparing}\mathrm{constant}\mathrm{term}\mathrm{on}$$$$\mathrm{each}\mathrm{side}:$$$$-1={(-1)}^{\mathrm{n}}{\alpha }^{1+2+....+(\mathrm{n}-1)}$$$$\Rightarrow 1.\alpha .{\alpha }^2.{\alpha }^3...{\alpha }^{\mathrm{n}-1}={(-1)}^{\mathrm{n}-1}.$$$$$$

Commented by Tinkutara last updated on 09/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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