Question

Questions :
$\\ \sf \: (1). \: Simplify : \: \frac{3 \sqrt{3} - 2 \sqrt{5}}{3 \sqrt{3} + 2 \sqrt{5}} + \dfrac{ \sqrt{12}}{ \sqrt{5} - 3 }$
$\sf \: (2). \: Simplify : \: \dfrac{2 \sqrt{3}}{ \sqrt{3} - \sqrt{2}} + \dfrac{3 \sqrt{3}}{ \sqrt{3} + \sqrt{2}}$




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Answer 1

Solution : 1

$\tt \implies \: \dfrac{3 \sqrt{3} - 2 \sqrt{5}}{3 \sqrt{3} + 2 \sqrt{5}} + \dfrac{ \sqrt{12}}{ \sqrt{5} - 3 }$

$\tt \implies \: \dfrac{3 \sqrt{3} - 2 \sqrt{5}}{3 \sqrt{3} + 2 \sqrt{5}} \times \dfrac{3 \sqrt{3} - 2 \sqrt{5}}{3 \sqrt{3} - 2 \sqrt{5}} + \dfrac{ 2\sqrt{3}}{ \sqrt{5} - 3 } \times \dfrac{ \sqrt{5} + 3 }{ \sqrt{5} + 3}$

$\tt \implies \: \dfrac{(3 \sqrt{3} )^{2} + (2 \sqrt{5})^{2} - 2 \times 3 \sqrt{3} \times 2 \sqrt{5}}{(3 \sqrt{3} )^{2} - (2 \sqrt{5})^{2}} + \dfrac{2 \sqrt{15}+ 6 \sqrt{3}}{( \sqrt{5})^{2} - 3 ^{2} }$

$\tt \implies \: \dfrac{27 + 20 - 12 \sqrt{15}}{27 - 20} + \dfrac{2( \sqrt{15} + 3 \sqrt{3} )}{5 - 9}$

$\tt \implies \: \dfrac{47- 12 \sqrt{15}}{7} - \dfrac{2( \sqrt{15} + 3 \sqrt{3} )}{4}$

$\tt \implies \: \dfrac{47- 12 \sqrt{15}}{7} - \dfrac{\sqrt{15} + 3 \sqrt{3} }{2}$

$\tt \implies \: \dfrac{94 - 24 \sqrt{15} - 7 \sqrt{15} - 21\sqrt{3}}{14}$

$\tt \: \implies \: \dfrac{94 - 31 \sqrt{15} - 21 \sqrt{3}}{14}$

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Solution : 2

$\tt \implies \: \dfrac{2 \sqrt{3}}{ \sqrt{3} - \sqrt{2}} + \dfrac{3 \sqrt{3}}{ \sqrt{3} + \sqrt{2}}$

$\tt \implies \: \dfrac{2 \sqrt{3}}{ \sqrt{3} - \sqrt{2}} \times \dfrac{ \sqrt{3} + \sqrt{3} }{ \sqrt{3} + \sqrt{2} } + \dfrac{3 \sqrt{3}}{ \sqrt{3} + \sqrt{2}} \times \dfrac{ \sqrt{3} - \sqrt{3} }{ \sqrt{3} - \sqrt{2} }$

$\tt \implies \: \dfrac{2 \sqrt{3}( \sqrt{3} + \sqrt{2}) } {( \sqrt{3} ) ^{2} - ( \sqrt{2} )^{2}} + \dfrac{3 \sqrt{3}( \sqrt{3} - \sqrt{2}) }{( \sqrt{3})^{2} - ( \sqrt{2})^{2}}$

$\tt \implies \: \dfrac{2 \times 3 \times2 \sqrt{6} }{3 - 2} + \dfrac{3 \times 3 - 3 \sqrt{6} }{3 - 2}$

$\tt \implies \: 6 + 2 \sqrt{6} + 9 - 3 \sqrt{6}$

$\tt \implies \: 15 - \sqrt{6}$