Question

If $x=8+5\sqrt{2}$, then find the value of $x+\frac{1}{x}$

Answer 1

If $x=8+5\sqrt{2}$ then

$x+\frac{1}{x}=8+5\sqrt{2}+\frac{1}{8+5\sqrt{2}}$

$x+\frac{1}{x}=8+5\sqrt{2}+\frac{1}{8+5\sqrt{2}}\times \frac{8-5\sqrt{2}}{8-5\sqrt{2}}$

$=8+5\sqrt{2}+\frac{8-5\sqrt{2}}{64-50}$

$=(8+\frac{8}{14})+(5\sqrt{2}-\frac{5\sqrt{2}}{14})=\frac{1}{14}(120+65\sqrt{2})$