Question

$\sqrt{2} + \sqrt{3} \div \sqrt{2} + \sqrt{3} + \sqrt{5}$
make the denominator rational

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Answer 1

$\underline{\underline{\blue{ \sf Question}}}$

• $\sf\dfrac{ \sqrt{2} + \sqrt{3} }{ \sqrt{2} + \sqrt{3} + \sqrt{5} }$

$\underline{\underline{\orange{ \sf Answer= \: 5 - \sqrt{10} - \sqrt{15}}}}$

$\underline{\underline{\pink{ \sf Step \ by \ step \ explanation}}}$

${\implies\sf\dfrac{ \sqrt{2} + \sqrt{3} }{ \sqrt{2} + \sqrt{3} + \sqrt{5} }}$

Rationalize the Term denominator

${\implies\sf\dfrac{ \sqrt{2} + \sqrt{3} }{ \sqrt{2} + \sqrt{3} + \sqrt{5} } \times \dfrac{ ((\sqrt{2} + \sqrt{3} ) - \sqrt{5} )}{ ((\sqrt{2} + \sqrt{3} ) - \sqrt{5} )}}$

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

${\implies \sf\dfrac{2 + \sqrt{6} - \sqrt{10} + \sqrt{6} + 3 - \sqrt{15} }{2 + \sqrt{6} - \sqrt{10} + \sqrt{6} + 3 - \sqrt{15} + \sqrt{10} + \sqrt{15} - 5 }}$

${\implies \sf\dfrac{5 + 2\sqrt{6} - \sqrt{10} - \sqrt{15} }{2 \sqrt{6} - \cancel{\sqrt{10}} + \sqrt{6} - \cancel{\sqrt{15}} + \cancel{\sqrt{10}} + \cancel{ \sqrt{15} } }}$

${ \implies\sf\dfrac{5 + \cancel{ 2\sqrt{6}} - \sqrt{10} - \sqrt{15} }{ \cancel{2 \sqrt{6} } } }$

$~~~~~~~ \blue{\sf \: 5 - \sqrt{10} - \sqrt{15}}$