Question

2/√5+√3
$\frac{2}{ \sqrt{5} + \sqrt{3 +} } + \frac{1}{ \sqrt{3} - \sqrt{2} } - \frac{3}{ \ \sqrt{5} + \sqrt{2} }$
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Answer 1

Rationalize

${ \sf\frac{2}{ \sqrt{5} + \sqrt{3 } } + \frac{1}{ \sqrt{3} - \sqrt{2} } - \frac{3}{ \sqrt{5} + \sqrt{2} }}$

$\underline{\mathsf{\green{One~ Rationalizing~ The~ Denominator}}}$

${ \implies\sf\frac{2}{ \sqrt{5} + \sqrt{3 } } \times \frac{ \sqrt{5 } - \sqrt{3} }{ \sqrt{5} - \sqrt{3} } + \frac{1}{ \sqrt{3} - \sqrt{2} } \times \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} + \sqrt{2} } - \frac{3}{ \ \sqrt{5} + \sqrt{2} } \times \frac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} - \sqrt{2} } }$

${\implies \sf\frac{2( \sqrt{5} - \sqrt{3}) }{ (\sqrt{5} )^{2} - ( \sqrt{3 })^{2} } + \frac{ \sqrt{3 } + \sqrt{2} }{ (\sqrt{3})^{2} - (\sqrt{2} )^{2} } - \frac{3( \sqrt{5} - \sqrt{2}) }{ ( \sqrt{5} )^{2} - ( \sqrt{2})^{2} }}$

${\implies \sf\frac{2 \sqrt{5} - 2\sqrt{3} }{5 - 3 } + \frac{ \sqrt{3 } + \sqrt{2} }{ 3 - 2 } - \frac{3 \sqrt{5} - 3 \sqrt{2}}{ 5 - 2 }}$

${\implies \sf\frac{2 \sqrt{5} - 2\sqrt{3} }{2 } + \frac{ \sqrt{3 } + \sqrt{2} }{ 1 } - \frac{3 \sqrt{5} - 3 \sqrt{2}}{ 3 }}$

Taking LCM of 2,1 and 3 =6

$\implies\sf \frac{3(2 \sqrt{5} - 2 \sqrt{3}) + 6 (\sqrt{3} + \sqrt{2}) - 2(3 \sqrt{5} - 3\sqrt{2}) }{6}$

$\implies\sf \frac{6 \sqrt{5} - 6 \sqrt{3} + 6 \sqrt{3} + 6 \sqrt{2} - 6 \sqrt{5} - 6\sqrt{2}}{6}$

$\implies\sf \frac{ \pink{ \cancel{6 \sqrt{5} }}- \green{ \cancel{6 \sqrt{3}}} + \green{\cancel{ 6 \sqrt{3}}} + \orange{\cancel{6 \sqrt{2}}} - \green{ 6 \sqrt{5}} - \orange{6\sqrt{2}}}{6}$

$~~~~~~~~~~~~~~~~~~~~~ \implies\sf \frac{0}{6}$

$~~~~~~~~~~~~~~~~~~~~~\implies\sf 0$

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Answer 2

Answer:

2√2

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