Question

If $\sqrt{5+2\sqrt{6}}=A+\sqrt{3}$ then $A=$ .

Answer 1

Given: $\sqrt{5+2\sqrt{6}}=A+\sqrt{3}$

Consider $\sqrt{5+2\sqrt{6}}$

$=\sqrt{5+2\times \sqrt{2}\times \sqrt{3}}\sqrt{6}\text{can be written as}\sqrt{2}\times \sqrt{3}$

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

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

Using $(a+b{)}^2=a^2+2ab+b^2$

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

$=\sqrt{3}+\sqrt{2}\dots \left(1\right)$

Now $\sqrt{5+2\sqrt{6}}=A+\sqrt{3}$

$\sqrt{3}+\sqrt{2}=A+\sqrt{3}\left(\text{from}1\right)$

Comparing we get,

$A=\sqrt{2}$