Question

solve it Pls no spam ​

Answer 1

$\bf{ \green{ \underline{ \underline{Solution}}}} \\ \\ \sf{ \frac{3 \sqrt{2} }{ \sqrt{6} - \sqrt{3} } + \frac{2 \sqrt{3} }{ \sqrt{6} + \sqrt{2} } - \frac{4 \sqrt{3} }{ \sqrt{6} - \sqrt{2} } }$

$\rm{ \pink{ \underline{1st\: part}}}\\ \\ \sf{ : \implies \frac{3 \sqrt{2} }{ \sqrt{6} - \sqrt{3} } \times \frac{ \sqrt{6} - \sqrt{3} }{ \sqrt{6} + \sqrt{3} } \: \: \: \:[Multiply\:by \: ( \sqrt{6} + \sqrt{3}) ] } \\ \\ \sf{ : \implies \frac{3 \sqrt{2} \: ( \sqrt{6} + \sqrt{3}) }{ { (\sqrt{6} )}^{2} - { (\sqrt{3} )}^{2} }} \\ \\ \sf{ : \implies \frac{6 \sqrt{3} + 3 \sqrt{6} }{2} } \\ \\ \sf{ : \implies \frac{ \cancel{3}(2 \sqrt{3} + \sqrt{6}) }{ \cancel{3} }} \\ \\ \sf{ : \implies 2 \sqrt{3} + \sqrt{6} }$

$\rm{ \pink{ \underline{2nd \: part}}} \\ \\ \sf{: \implies \frac{2 \sqrt{3} }{ \sqrt{6} + 2 } \times \frac{ \sqrt{6} - 2}{ \sqrt{6} - 2} \: \: \: \: [Multiply\:by \: ( \sqrt{6} - 2)]} \\ \\ \sf{: \implies \frac{2 \sqrt{3} \: ( \sqrt{6} - 2)}{ (\sqrt{6}) ^{2} - {(4)}^{2} } } \\ \\ \sf{: \implies \frac{6 \sqrt{2} - 4 \sqrt{3} }{6 - 4} } \\ \\ \sf{: \implies \frac{ \cancel{2 }\: (3 \sqrt{2} - 2 \sqrt{3})}{ \cancel{2} }} \\ \\ \sf{: \implies 3 \sqrt{2 } - 2 \sqrt{3} }$

$\rm{ \pink{ \underline{3rd \: part}}} \\ \\ \sf{: \implies \frac{4 \sqrt{3} }{ \sqrt{6} - \sqrt{2}} \times \frac{ \sqrt{6} + \sqrt{2} }{ \sqrt{6} + \sqrt{2} } } \: \: \: \: [Multiply \: by \: ( \sqrt{6} + \sqrt{2}) ] \\ \\ \sf{: \implies \frac{4 \sqrt{3} \: ( \sqrt{6} + \sqrt{2} )}{ ({ \sqrt{6}) }^{2} - { (\sqrt{2} }^{2} )} } \\ \\ \sf{: \implies \frac{12 \sqrt{2} + 4 \sqrt{6} }{6 - 2} } \\ \\ \sf{: \implies \frac{ \cancel{4}\: ( \sqrt{2} + \sqrt{6}) }{ \cancel{4} }} \\ \\ \sf{ : \implies 3 \sqrt{2} + \sqrt{6} }$

$\rm{ \orange{Now,}} \\ \\ \sf{ \frac{3 \sqrt{2} }{ \sqrt{6} - \sqrt{3} } + \frac{2 \sqrt{3} }{ \sqrt{6} + \sqrt{2} } - \frac{4 \sqrt{3} }{ \sqrt{6} - \sqrt{2} } } \\ \\ \sf{ : \implies (\cancel{2{ \sqrt{3} }} + \cancel{\sqrt{6}} ) + (\cancel{3 \sqrt{2} } - \cancel{2 \sqrt{3}}) - (\cancel{3 \sqrt{2} } - \cancel{ \sqrt{6} }}) \\ \\ \sf{ \purple{ : \implies \underline{ \boxed{ \sf{0}}} }}$