If-x-26-521x-13-1-x-10-1-x-10-Any-help-

Question Number 138469 by KwesiDerek last updated on 13/Apr/21

If x^{26} - 521 x^{13} = 1

x^{10} + \frac{1}{x^{10}} = ?

Any help

Answered by MJS_new last updated on 14/Apr/21

x^{13} = \frac{521 \pm 233 \sqrt{5}}{2}

just trying around, at the end approximating

\sqrt[13]{\frac{521 + 233 \sqrt{5}}{2}} \approx 1.618034 \Rightarrow

x = \frac{1 \pm \sqrt{5}}{2}

\Rightarrow x^{10} + \frac{1}{x^{10}} = 123

Answered by TheSupreme last updated on 14/Apr/21

x^{13} = t

t^{2} - 521 t + 1 = 0 \to t = \frac{521 \pm \sqrt{521^{2} - 4}}{2}

x^{10} = t^{\frac{10}{13}}

x^{10} + \frac{1}{x^{10}} = {[\frac{521 \pm \sqrt{521^{2} - 4}}{2}]}^{\frac{10}{13}} + {[\frac{2}{521 \pm \sqrt{521^{2} - 4}}]}^{\frac{10}{13}}

Commented by Rasheed.Sindhi last updated on 14/Apr/21

T y p o s i r . t^{2} - 521 t - 1 = 0