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Question Number 163631 by HongKing last updated on 08/Jan/22

$$\mathrm{x}+\frac{1}{\mathrm{x}}=\sqrt{3}\Rightarrow {\mathrm{x}}^{3579}+\frac{1}{{\mathrm{x}}^{3579}}=?$$

Answered by MathsFan last updated on 09/Jan/22

$${\mathrm{x}}^2-\mathrm{x}\sqrt{3}+1=0\to \mathrm{x}=\frac{i\pm \sqrt{3}}{2}$$$${\left(\frac{i\pm \sqrt{3}}{2}\right)}^{3579}+\frac{1}{{\left(\frac{i\pm \sqrt{3}}{2}\right)}^{3579}}=0$$

Commented by behi834171 last updated on 09/Jan/22

$$x=\frac{\sqrt{3}}{2}+\frac{1}{2}i=cos\frac{\pi }{6}+isin\frac{\pi }{6}=e^{i\frac{\pi }{6}}\Rightarrow$$$$x^{3679}+\frac{1}{x^{3579}}=e^{3579\times \frac{i\pi }{6}}=cos\left(\frac{1193\pi }{2}\right)+isin\left(\frac{1193\pi }{2}\right)=$$$$=cos(596\pi +\frac{\pi }{2})+isin(596\pi +\frac{\pi }{2})=\mathit{i}.◼$$

Answered by mr W last updated on 09/Jan/22

$$x^2-\sqrt{3}x+1=0$$$$x=\frac{\sqrt{3}\pm i}{2}=\cos (\pm \frac{\pi }{6})+i\sin (\pm \frac{\pi }{6})$$$$$$$$x^{3579}+\frac{1}{x^{3579}}=\cos (\pm \frac{3579\pi }{6})+i\sin (\pm \frac{3579\pi }{6})+\cos (\mp \frac{3579\pi }{6})+i\sin (\mp \frac{3579\pi }{6})$$$$x^{3579}+\frac{1}{x^{3579}}=\cos (\pm \frac{3579\pi }{6})+i\sin (\pm \frac{3579\pi }{6})+\cos (\pm \frac{3579\pi }{6})-i\sin (\pm \frac{3579\pi }{6})$$$$x^{3579}+\frac{1}{x^{3579}}=2\cos \left(\frac{3579\pi }{6}\right)$$$$x^{3579}+\frac{1}{x^{3579}}=2\cos (298\times 2\pi +\frac{\pi }{2})$$$$x^{3579}+\frac{1}{x^{3579}}=2\cos \left(\frac{\pi }{2}\right)$$$$x^{3579}+\frac{1}{x^{3579}}=2\times 0$$$$x^{3579}+\frac{1}{x^{3579}}=0$$

Commented by HongKing last updated on 09/Jan/22

$$\mathrm{cool}\mathrm{my}\mathrm{dear}\mathrm{Sir}\mathrm{thank}\mathrm{you}\mathrm{so}\mathrm{much}$$

Commented by peter frank last updated on 11/Jan/22

$$\mathrm{thank}\mathrm{you}$$

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