The simplified value of the following expression is:
$$\frac{1}{{\sqrt {11 - 2\sqrt {30} } }} - \frac{3}{{\sqrt {7 - 2\sqrt {10} } }} - \frac{4}{{\sqrt {8 + 4\sqrt 3 } }}$$

Solution (By Examveda Team)

$$\eqalign{ & \frac{1}{{\sqrt {11 - 2\sqrt {30} } }} \cr & = \frac{1}{{\sqrt {6 + 5 - 2 \times \sqrt 6 \times \sqrt 5 } }} \cr & = \frac{1}{{\sqrt {{{\left( {\sqrt 6 } \right)}^2} + {{\left( {\sqrt 5 } \right)}^2} - 2 \times \sqrt 6 \times \sqrt 5 } }} \cr & = \frac{1}{{\sqrt {{{\left( {\sqrt 6 - \sqrt 5 } \right)}^2}} }} \cr & = \frac{1}{{\sqrt 6 - \sqrt 5 }} \cr & = \frac{{\left( {\sqrt 6 + \sqrt 5 } \right)}}{{\left( {\sqrt 6 - \sqrt 5 } \right)\left( {\sqrt 6 + \sqrt 5 } \right)}} \cr & = \sqrt 6 + \sqrt 5 \cr & \frac{3}{{\sqrt {7 - 2\sqrt {10} } }} \cr & = \frac{3}{{\sqrt {5 + 2 - 2 \times \sqrt 5 \times \sqrt 2 } }} \cr & = \frac{3}{{\sqrt 5 - \sqrt 2 }} \cr & = \frac{{3 \times \left( {\sqrt 5 + \sqrt 2 } \right)}}{{\left( {\sqrt 5 - \sqrt 2 } \right)\left( {\sqrt 5 + \sqrt 2 } \right)}} \cr & = \frac{{3\left( {\sqrt 5 + \sqrt 2 } \right)}}{{5 - 2}} \cr & = \sqrt 5 + \sqrt 2 \cr & \frac{4}{{\sqrt {8 + 4\sqrt 3 } }} \cr & = \frac{4}{{\sqrt {8 + 2\sqrt {12} } }} \cr & = \frac{4}{{\sqrt {6 + 2 + 2 \times \sqrt 6 \times \sqrt 2 } }} \cr & = \frac{4}{{\sqrt {{{\left( {\sqrt 6 + \sqrt 2 } \right)}^2}} }} \cr & = \frac{{4 \times \left( {\sqrt 6 - \sqrt 2 } \right)}}{{\left( {\sqrt 6 + \sqrt 2 } \right)\left( {\sqrt 6 - \sqrt 2 } \right)}} \cr & = \frac{{4\left( {\sqrt 6 - \sqrt 2 } \right)}}{{6 - 2}} \cr & = \sqrt 6 - \sqrt 2 \cr & \therefore {\text{Expression}} \cr & = \left( {\sqrt 6 + \sqrt 5 } \right) - \left( {\sqrt 5 + \sqrt 2 } \right) - \left( {\sqrt 6 - \sqrt 2 } \right) \cr & = \sqrt 6 + \sqrt 5 - \sqrt 5 - \sqrt 2 - \sqrt 6 + \sqrt 2 \cr & = 0 \cr} $$