Question

$\sqrt{16+2\sqrt{6}+2\sqrt{22}+2\sqrt{33}}=$?

Answer 1

We have

$\sqrt{16+2\sqrt{6}+2\sqrt{22}+3\sqrt{33}}$

$=\sqrt{2+3+11+2.\sqrt{2}.\sqrt{3}+2\sqrt{2}.\sqrt{11}+2.\sqrt{3}.\sqrt{11}}$

$=\sqrt{{\left(\sqrt{2}\right)}^2+{\left(\sqrt{3}\right)}^2+{\left(\sqrt{11}\right)}^2+2.\sqrt{2}.\sqrt{3}+2.\sqrt{2}.\sqrt{11}+2.\sqrt{11}.\sqrt{2}}$

$=\sqrt{{\left(\sqrt{2}+\sqrt{3}+\sqrt{11}\right)}^2}$

$=\sqrt{2}+\sqrt{3}+\sqrt{11}$, which is the required answer.