Question

Express
$\frac{1 + \sqrt{2} }{ \sqrt{5} + \sqrt{3} } + \frac{1 - \sqrt{2} }{ \sqrt{5} - \sqrt{3} }$
in the form
$x \sqrt{5} + y \sqrt{6}$
​

Answer 1

$\: \: \: \: \frac{1 + \sqrt{2} }{ \sqrt{5} + \sqrt{3} } + \frac{1 - \sqrt{2} }{ \sqrt{5} - \sqrt{3} } \\ = \frac{(1 - \sqrt{2})( \sqrt{5} - \sqrt{3}) + (1 - \sqrt{2} )( \sqrt{5} + \sqrt{3}) }{( \sqrt{5} + \sqrt{3})( \sqrt{5} - \sqrt{3}) } \\ = \frac{ \sqrt{5} - \sqrt{3} - \sqrt{10} + \sqrt{6} + \sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6} }{ {( \sqrt{5}) }^{2} - {( \sqrt{3} )}^{2} } \\ \\ (swipe\: left \:because \:answer\: is\: continued \: \\towards\: right) \\ \\ = \frac{2 \sqrt{5} - 2 \sqrt{10} }{5 - 3} \\ = \frac{2( \sqrt{5} - \sqrt{10} )}{2} \\ = \sqrt{5} - \sqrt{10} \\ = \sqrt{5} + ( - \frac{ \sqrt{10} \sqrt{6} }{ \sqrt{6} } ) \\ = \sqrt{5} + ( - \sqrt{ \frac{10}{6} }) \sqrt{6} \\ = (1) \sqrt{5} + ( - \sqrt{ \frac{5}{3} } ) \sqrt{6}$

Hence, the given equation can be expressed as x√5 + y√6 when x = 1 and y = -√(5/3)