Question

If $x=11+6\sqrt{2},$ then the value of $\sqrt{x}-\frac{1}{\sqrt{x}}$ is

• A. $\frac{2}{7}\left(4\sqrt{2}-9\right)$
• B. $\frac{2}{7}\left(9+4\sqrt{2}\right)$
• C. $\frac{2}{7}\left(4+9\sqrt{2}\right)$
• D. $\frac{2}{7}\left(9\sqrt{2}-4\right)$

Answer 1

The correct option is B $\frac{2}{7}\left(9+4\sqrt{2}\right)$

Given,

$x=11+6\sqrt{2},$

or, $x=3^2+(\sqrt{2}{)}^2+\mathrm{2.3.}\sqrt{2}$

or, $x=(3+\sqrt{2}{)}^2$

or, $\sqrt{x}=3+\sqrt{2}$.

Then

$\sqrt{x}-\frac{1}{\sqrt{x}}$

$=3+\sqrt{2}-\frac{1}{3+\sqrt{2}}$

$=3+\sqrt{2}-\frac{(3-\sqrt{2})}{7}$

$=\frac{18}{7}+\frac{8\sqrt{2}}{7}$

$=\frac{2}{7}(9+4\sqrt{2})$.