The value of $$\frac{{3\sqrt 7 }}{{\sqrt 5 + \sqrt 2 }} - \frac{{5\sqrt 5 }}{{\sqrt 2 + \sqrt 7 }} + \frac{{2\sqrt 2 }}{{\sqrt 7 + \sqrt 5 }}$$       is:

Solution (By Examveda Team)

$$\eqalign{ & \frac{{3\sqrt 7 }}{{\sqrt 5 + \sqrt 2 }} - \frac{{5\sqrt 5 }}{{\sqrt 2 + \sqrt 7 }} + \frac{{2\sqrt 2 }}{{\sqrt 7 + \sqrt 5 }} \cr & = \frac{{3\sqrt 7 }}{{\sqrt 5 + \sqrt 2 }} \times \frac{{\sqrt 5 - \sqrt 2 }}{{\sqrt 5 - \sqrt 2 }} - \frac{{5\sqrt 5 }}{{\sqrt 7 + \sqrt 2 }} \times \frac{{\sqrt 7 - \sqrt 2 }}{{\sqrt 7 - \sqrt 2 }} + \frac{{2\sqrt 2 }}{{\sqrt 7 + \sqrt 5 }} \times \frac{{\sqrt 7 - \sqrt 5 }}{{\sqrt 7 - \sqrt 5 }} \cr & = \frac{{3\sqrt 7 \left( {\sqrt 5 - \sqrt 2 } \right)}}{{{{\left( {\sqrt 5 } \right)}^2} - {{\left( {\sqrt 2 } \right)}^2}}} - \frac{{5\sqrt 5 \left( {\sqrt 7 - \sqrt 2 } \right)}}{{{{\left( {\sqrt 7 } \right)}^2} - {{\left( {\sqrt 2 } \right)}^2}}} + \frac{{2\sqrt 2 \left( {\sqrt 7 - \sqrt 5 } \right)}}{{{{\left( {\sqrt 7 } \right)}^2} - {{\left( {\sqrt 5 } \right)}^2}}} \cr & = \sqrt {35} - \sqrt {14} - \sqrt {35} + \sqrt {10} + \sqrt {14} - \sqrt {10} \cr & = 0 \cr} $$