Question

Rationalise it
(root 2 - 3)^-1 + (root 2 + root 3 ) ^-1

Answer 1

${( \sqrt{2} - 3)}^{ - 1} + ( { \sqrt{2} + \sqrt{3}) }^{ - 1} \\ = \frac{1}{ \sqrt{2} - 3 } + \frac{1}{ \sqrt{2} + \sqrt{3} } \\ = \frac{1( \sqrt{2} + 3) }{ (\sqrt{2} - 3)( \sqrt{2} + 3) } + \frac{1( \sqrt{2} - \sqrt{3} )}{ (\sqrt{2} + \sqrt{3})( \sqrt{2} - \sqrt{3} ) } \\ = \frac{( \sqrt{2} + 3) }{2 - 9} + \frac{( \sqrt{2} - \sqrt{3}) }{2 - 3} \\ = \frac{( \sqrt{2} + 3) }{ - 7} + \frac{( \sqrt{2} - \sqrt{3}) }{ - 1} \\ = \frac{ - ( \sqrt{2} + 3) - 7( \sqrt{2} - \sqrt{3}) }{7} \\ = \frac{ - \sqrt{2} - 3 - 7 \sqrt{2} + \sqrt{3} }{7} \\ = \frac{ - 8 \sqrt{2} - 3 + \sqrt{3} }{7}$