Evaluate frac 1 sqrt 2 sqrt 5 sqrt 3 frac 1 - s... - SS3 Mathematics Surd Question

\(\frac{1 + \sqrt{2}}{\sqrt{5} + \sqrt{3}} + \frac{1 - \sqrt{2}}{\sqrt{5} - \sqrt{3}} = \ \frac{\left( \sqrt{5} - \sqrt{3} \right)\left( 1 + \sqrt{2} \right) + (\sqrt{5} + \sqrt{3})(1 - \sqrt{2})}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})} = \frac{\left( \sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6} \right) + (\sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6})}{5 - 3}\)

\(\frac{\sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6} + \sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}}{2} = \ \frac{2\sqrt{5} - 2\sqrt{6}}{2} = \frac{2(\sqrt{5} - \sqrt{6})}{2} = \sqrt{5} - \sqrt{6}\)