Question

If $x=\sqrt{3+2\sqrt{2}}$, then find the value of $x^4+\frac{1}{x^4}$

• A. $34$
• B. $35$
• C. $36$
• D. $37$

Answer 1

The correct option is A $34$

$x^4+\frac{1}{x^4}={(x^2+\frac{1}{x^2})}^2-2$

$\Rightarrow {\left[\right(3+2\sqrt{2})+\frac{1}{(3+2\sqrt{2})}]}^2-2$

$\Rightarrow {[3+2\sqrt{2}+\frac{(3-2\sqrt{2})}{3^2-(2\sqrt{2}{)}^2}]}^2-2$

$\Rightarrow \{{[3+2\sqrt{2}+3-2\sqrt{2}]}^2-2\}=34$