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

$$\begin{gathered} 5+2\sqrt{6}=2+3+2\times \sqrt{3}\times \sqrt{2} \\ ={\left(\sqrt{2}\right)}^2+{\left(\sqrt{3}\right)}^2+2\times \sqrt{3}\times \sqrt{2} \end{gathered}$$
This is in the form
$a^2+b^2+2ab={\left(a+b\right)}^2$
So, we have
${\left(\sqrt{2}\right)}^2+{\left(\sqrt{3}\right)}^2+2\times \sqrt{3}\times \sqrt{2}={\left(\sqrt{2}+\sqrt{3}\right)}^2$
Thus, $\sqrt{5+2\sqrt{6}}=\sqrt{{\left(\sqrt{2}+\sqrt{3}\right)}^2}=\sqrt{2}+\sqrt{3}$
Hence, the correct answer is option (c).