Question

Find the square root of
$5-\sqrt{10}-\sqrt{15}+\sqrt{6}$

Answer 1

$5-\sqrt{10}-\sqrt{15}+\sqrt{6}$

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

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

$=(1+\sqrt{\frac{3}{2}}-\sqrt{\frac{5}{2}}{)}^2$

Hence square root of $5-\sqrt{10}-\sqrt{15}+\sqrt{6}$ is $=\pm (1+\sqrt{\frac{3}{2}}-\sqrt{\frac{5}{2}})$