If $$\frac{{22\sqrt 2 }}{{4\sqrt 2 - \sqrt {3 + \sqrt 5 } }} = a + \sqrt 5 b$$      with a, b > 0, then what is the value of (ab) : (a + b)?

Solution (By Examveda Team)

$$\eqalign{ & \frac{{22\sqrt 2 }}{{4\sqrt 2 - \sqrt {3 + \sqrt 5 } }} = a + \sqrt 5 b \cr & {\text{LHS}} = \frac{{22\sqrt 2 }}{{4\sqrt 2 - \sqrt {3 + \sqrt 5 } }} \cr & = \frac{{22\sqrt 2 }}{{4\sqrt 2 - \sqrt {\frac{{6 + 2\sqrt 5 }}{2}} }} \cr & = \frac{{22\sqrt 2 }}{{4\sqrt 2 - \frac{{\sqrt 5 + 1}}{{\sqrt 2 }}}} \cr & = \frac{{22\sqrt 2 }}{{\frac{{8 - \sqrt 5 - 1}}{{\sqrt 2 }}}} \cr & = \frac{{22 \times 2}}{{7 - \sqrt 5 }} \times \frac{{7 + \sqrt 5 }}{{7 + \sqrt 5 }} \cr & = \frac{{44\left( {7 + \sqrt 5 } \right)}}{{{7^2} - {{\left( {\sqrt 5 } \right)}^2}}} \cr & = \frac{{44\left( {7 + \sqrt 5 } \right)}}{{44}} \cr & = 7 + \sqrt 5 \cr & {\text{Compare LHS and RHS}} \cr & 7 + \sqrt 5 = a + \sqrt 5 b \cr & a = 7;\,b = 1 \cr & \left( {ab} \right):\left( {a + b} \right) = \left( {7 \times 1} \right):\left( {7 + 1} \right) = 7:8 \cr} $$