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Answer 1

Question -

Simplify -

$( { \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} }) }^{2} + ({ \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} }) }^{2}$

Solution-

$= ( { \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } )}^{2} + ({ \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } )}^{2}$

$= [ { \frac{( \sqrt{3} - \sqrt{2})( \sqrt{3} - \sqrt{2} )}{( \sqrt{3} + \sqrt{2})( \sqrt{3} - \sqrt{2}) } } ]^{2} + [ { \frac{( \sqrt{3} + \sqrt{2} )( \sqrt{3} + \sqrt{2}) }{( \sqrt{3} - \sqrt{2} )( \sqrt{3} + \sqrt{2} } } ]^{2}$

$= [ { \frac{ {( \sqrt{3}) }^{2} + {( \sqrt{2}) }^{2} - 2 \sqrt{6} }{ {( \sqrt{3}) }^{2} - {( \sqrt{2} )}^{2} } } ]^{2} + [ { \frac{ { \sqrt{3} }^{2} + { \sqrt{2} }^{2} + 2 \sqrt{6} }{ {( \sqrt{3}) }^{2} - {( \sqrt{2} )}^{2} } } ]^{2}$

$= (\frac{3 + 2 - 2 \sqrt{6} }{3 - 2} ) {}^{2} +( \frac{3 + 2 + 2 \sqrt{6} }{3 - 2} ) {}^{2}$

$= {(5 - 2 \sqrt{6} )}^{2} + {(5 + 2 \sqrt{6}) }^{2}$

$= {(5)}^{2} + {(2 \sqrt{6}) }^{2} - 2 \times 5 \times 2 \sqrt{6} + {(5)}^{2} + {(2 \sqrt{6}) }^{2} + 2 \times 5 \times 2 \sqrt{6}$

$= 25 + 24 + 25 + 24$

$= 49 + 49$

$= 98$