If x, y are rational numbers and$$\frac{{5 + \sqrt {11} }}{{3 - 2\sqrt {11} }} = x + y\sqrt {11} .$$     The values of x and y are

$$\eqalign{ & \frac{{5 + \sqrt {11} }}{{3 - 2\sqrt {11} }} = x + y\sqrt {11} \cr & \frac{{\left( {5 + \sqrt {11} } \right)\left( {3 + 2\sqrt {11} } \right)}}{{\left( {3 - 2\sqrt {11} } \right)\left( {3 + 2\sqrt {11} } \right)}} = x + y\sqrt {11} \cr & \frac{{15 + 10\sqrt {11} + 3\sqrt {11} + 22}}{{9 - 44}} = x + y\sqrt {11} \cr & \frac{{37 - 13\sqrt {11} }}{{ - 35}} = x + y\sqrt {11} \cr & \frac{{ - 37}}{{35}} + \frac{{\left( { - 13} \right)\sqrt {11} }}{{35}} = x + y\sqrt {11} \cr & {\text{Comparing both side}} \cr & x = \frac{{ - 37}}{{35}} \cr & y = \frac{{ - 13}}{{35}} \cr} $$