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

Explanation:-

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