Question

Find the value of
$\frac{\sqrt{26-15\sqrt{3}}}{5\sqrt{2}-\sqrt{38+5\sqrt{3}}}$

Answer 1

$\frac{\sqrt{26-15\sqrt{3}}}{5\sqrt{2}-\sqrt{38+5\sqrt{3}}}$

$=\frac{\sqrt{52-30\sqrt{3}}}{\sqrt{2}(5\sqrt{2}-\sqrt{38+5\sqrt{3}})}$ ..... Multiplying and dividing by $\sqrt{2}$

$=\frac{\sqrt{(3\sqrt{3}-5{)}^2}}{\sqrt{2}(5\sqrt{2}-\sqrt{38+5\sqrt{3}})}$

$=\frac{3\sqrt{3}-5}{\sqrt{2}\times (5\sqrt{2}-\sqrt{38+5\sqrt{3}})}$

$=\frac{3\sqrt{3}-5}{10-\sqrt{76+10\sqrt{3}}}$

$=\frac{3\sqrt{3}-5}{10-(1+5\sqrt{3})}$

$=\frac{3\sqrt{3}-5}{9-\sqrt{3}}$

$=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$