Question

If $x=9-4\sqrt{5}$ then, find the value of $x^6+\frac{1}{x^6}$

Answer 1

We have,

$\begin{array}{c}x=9-4\sqrt{5}\\ \frac{1}{x}=9+4\sqrt{5}\\ x^2+\frac{1}{x^2}=322\\ x^6+\frac{1}{x^6}+3\left(322\right)={\left(322\right)}^3\\ x^6+\frac{1}{x^6}=33386248-966\\ x^6+\frac{1}{x^6}=33,385,282\end{array}$

Hence, this is the answer.