Question

Please tell the 7 question this is too important please

Answer 1

$\bf \: \frac{ 7\sqrt{3} }{ \sqrt{10} + \sqrt{3} } - \frac{2 \sqrt{5} }{ \sqrt{6} + \sqrt{5} } - \frac{3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} }$

We solution this question in 3 parts ;

$\tt \: First \: part \: \\ \sf \: \frac{7 \sqrt{3} }{ \sqrt{10} + \sqrt{3} }$

$\tt \implies \: \frac{7 \sqrt{3} }{ \sqrt{10} + \sqrt{3} } \times \frac{ \sqrt{10} - \sqrt{3} }{ \sqrt{10} - \sqrt{3} }$

$\sf \implies \frac{7 \sqrt{3}( \sqrt{10} - \sqrt{3} ) }{ {( \sqrt{10}) }^{2} - {( \sqrt{3} )}^{2} }$

$\sf \implies \frac{ \cancel7 \sqrt{3} ( \sqrt{10} - \sqrt{3} }{ \cancel7}$

$\sf \implies \sqrt{3} ( \sqrt{10} - \sqrt{3} )$

$\sf \implies \sqrt{30} - 3$

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$\tt \: Second \: part \\ \sf \: \frac{2 \sqrt{5} }{ \sqrt{6} + \sqrt{3} }$

$\sf \implies \frac{2 \sqrt{5} }{ \sqrt{6} + \sqrt{5} } \times \frac{ \sqrt{6} - \sqrt{5} }{ \sqrt{6} - \sqrt{5} }$

$\sf \implies \frac{2 \sqrt{5}( \sqrt{6} - \sqrt{5} ) }{ {( \sqrt{6} )}^{2} - {( \sqrt{5}) }^{2} }$

$\sf \implies \frac{2 \sqrt{5}( \sqrt{6} - \sqrt{5} )}{1}$

$\sf \implies2 \sqrt{5} ( \sqrt{6} - \sqrt{5} )$

$\sf \implies2 \sqrt{30} - 10$

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$\tt \: Third \: part \\ \sf \: \frac{3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} }$

$\sf \implies \frac{3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} } \times \frac{ \sqrt{15} - 2\sqrt{2} }{ \sqrt{15} - 3 \sqrt{2} }$

$\sf \implies \frac{3 \sqrt{2} ( \sqrt{15} - 3 \sqrt{2} ) }{ {(15)}^{2} - {(3 \sqrt{2} )}^{2} }$

$\sf \implies \frac{3 \sqrt{2}( \sqrt{15} - 3 \sqrt{2} )}{15 - 9 \times 2}$

$\sf \implies \frac{3 \sqrt{2}( \sqrt{15} - 3 \sqrt{2}) }{ - 3}$

$\sf \implies - \sqrt{2} ( \sqrt{15} - 3 \sqrt{2} )$

$\sf \implies - \sqrt{30} + 6$

Now,

$\bf \: \frac{ 7\sqrt{3} }{ \sqrt{10} + \sqrt{3} } - \frac{2 \sqrt{5} }{ \sqrt{6} + \sqrt{5} } - \frac{3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} }$

$\implies \tt \: ( \sqrt{30} - 3) - 2 \sqrt{30} - 10 - ( - \sqrt{30} + 6)$

$\tt \implies \sqrt{30} + 3 - 2 \sqrt{30} + 10 + \sqrt{30} - 6$

$\tt \implies( \sqrt{30} - 2 \sqrt{30} + \sqrt{30} ) + ( - 3 + 10 - 6)$

$\tt \implies \: 1$