The simplified form of $$\frac{2}{{\sqrt 7 + \sqrt 5 }} + $$   $$\frac{7}{{\sqrt {12} - \sqrt 5 }} - $$   $$\frac{5}{{\sqrt {12} - \sqrt 7 }}$$   is = ?

Solution (By Examveda Team)

$$\frac{2}{{\sqrt 7 + \sqrt 5 }} + \frac{7}{{\sqrt {12} - \sqrt 5 }} - \frac{5}{{\sqrt {12} - \sqrt 7 }}$$
$$ = \frac{2}{{\sqrt 7 + \sqrt 5 }} \times $$  $$\frac{{\sqrt 7 - \sqrt 5 }}{{\sqrt 7 - \sqrt 5 }} + $$  $$\frac{7}{{\sqrt {12} - \sqrt 5 }} \times $$  $$\frac{{\sqrt {12} + \sqrt 5 }}{{\sqrt {12} + \sqrt 5 }} - $$  $$\left( {\frac{5}{{\sqrt {12} - \sqrt 7 }} \times \frac{{\sqrt {12} + \sqrt 7 }}{{\sqrt {12} + \sqrt 7 }}} \right)$$
$$ = \frac{{2\left( {\sqrt 7 - \sqrt 5 } \right)}}{2} + $$   $$\frac{{7\left( {\sqrt {12} + \sqrt 5 } \right)}}{7} - $$   $$\frac{{5\left( {\sqrt {12} + \sqrt 7 } \right)}}{5}$$
$$\eqalign{ &= \sqrt 7 - \sqrt 5 + \sqrt {12} + \sqrt 5 - \sqrt {12} - \sqrt 7 \cr &= 0 \cr} $$