Question

The value of $\sqrt{5+2\sqrt{6}}$ is

(a)$\sqrt{5}+\sqrt{6}$ (b)$\sqrt{5}-\sqrt{6}$ (c)$\sqrt{3}+\sqrt{2}$ (d)$\sqrt{3}-\sqrt{2}$

Answer 1

$5+2\sqrt{6}=2+3+2\times \sqrt{2}\times \sqrt{3}$

This is of the form of $a^2+b^2+2ab=(a+b{)}^2$

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

So, $\sqrt{5+2\sqrt{6}}=\sqrt{(\sqrt{2}+\sqrt{3}{)}^2}=\sqrt{2}+\sqrt{3}$

Hence, the correct answer is the option (c)