Question

If $x=8+2\sqrt{15},$ then the value of $\sqrt{x}+\frac{1}{\sqrt{x}}$ is

• A. $2\sqrt{3}$
• B. $2\sqrt{5}$
• C. $\frac{3}{2}\sqrt{5}+\frac{\sqrt{3}}{2}$
• D. $\frac{\sqrt{5}}{2}+\frac{3}{2}\sqrt{3}$

Answer 1

The correct option is C $\frac{3}{2}\sqrt{5}+\frac{\sqrt{3}}{2}$

$\sqrt{x}=\sqrt{8+2\sqrt{15}}$

$=\sqrt{5+3+2\sqrt{5\times 3}}$
$=\sqrt{(\sqrt{5}{)}^2+(\sqrt{3}{)}^2+2\times \sqrt{5}\times \sqrt{3}}$
$=\sqrt{(\sqrt{5}+\sqrt{3}{)}^2}$

$=\sqrt{5}+\sqrt{3}$

$\therefore \sqrt{x}+\frac{1}{\sqrt{x}}=(\sqrt{5}+\sqrt{3})+\frac{1}{(\sqrt{5}+\sqrt{3})}$
$=(\sqrt{5}+\sqrt{3})+\frac{\sqrt{5}-\sqrt{3}}{(\sqrt{5}+\sqrt{3})\left(\right(\sqrt{5}-\sqrt{3})}$
$=(\sqrt{5}+\sqrt{3})+\frac{(\sqrt{5}-\sqrt{3})}{5-3}$
$=(\sqrt{5}+\sqrt{3})+\frac{(\sqrt{5}-\sqrt{3})}{2}$
$=\frac{2\sqrt{5}+2\sqrt{3}+\sqrt{5}-\sqrt{3}}{2}$
$=\frac{3\sqrt{5}}{2}+\frac{\sqrt{3}}{2}$