Question

Simplify
$\frac{6}{\sqrt{24-2\sqrt{135}}}-\sqrt{15}$

Answer 1

$135=3^3\times 5=9\times 15$

$24=15+9$

$24-2\times \sqrt{135}=15+9-2\times \sqrt{9\times 15}=(\sqrt{15}-3{)}^2$

$\frac{6}{\sqrt{(\sqrt{15}-3{)}^2}}-\sqrt{15}=\frac{6}{\sqrt{15}-3}-\sqrt{15}$

$=\frac{6\times (\sqrt{15}+3)}{(\sqrt{15}-3)\times (\sqrt{15}+3)}-\sqrt{15}=\frac{6\times (\sqrt{15}+3)}{6}-\sqrt{15}=3$