Question

$~~~~~~~~$${\bf{Simplify}}$ -
$~$
$\frac{3}{ \sqrt{5} + \sqrt{2} } - \frac{2}{ \sqrt{7} + \sqrt{5} } + \frac{5}{ \sqrt{7} - \sqrt{2} }$
${\bf\pink{All \: The \: Best !! \: ♡}}$​

Answer 1

$\large {\textsf {\textbf \purple {✿ \: Question \:➯ }}}$

Simplify the following :-

$\sf \cfrac{3}{ \sqrt{5} + \sqrt{2} } - \cfrac{2}{ \sqrt{7} + \sqrt{5} } + \cfrac{5}{ \sqrt{7} - \sqrt{2} }$

$\large {\textsf {\textbf \purple {✿ \: Solution \: ➯}}}$

Rationalising the denominators :-

$\blue{(1) ➯} \: \sf \cfrac{3}{ \sqrt{5} + \sqrt{2} } \times ( \cfrac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} - \sqrt{2} } )$

$\blue ➯ \: \sf \cfrac{3( \sqrt{5} - \sqrt{2}) }{ { \sqrt{5} }^{2} - { \sqrt{2} }^{2} } = \cfrac {\cancel3 (\sqrt{5} - \sqrt{2} )}{\cancel3}$

$\underline {\boxed {\blue ➯ \: \sf \cfrac{3}{ \sqrt{5} - \sqrt{2} } = \sqrt{5} - \sqrt{2} }}$

$\blue {(2)➯} \: \sf \cfrac{2}{ \sqrt{7} + \sqrt{5} } \times ( \cfrac{ \sqrt{7} - \sqrt{5} }{ \sqrt{7} - \sqrt{5} } )$

$\blue ➯ \: \sf \cfrac{2( \sqrt{7} - \sqrt{5} )}{ { \sqrt{7} }^{2} - { \sqrt{5} }^{2} } = \cfrac{ \cancel2( \sqrt{7} - \sqrt{5}) }{ \cancel2}$

$\underline {\boxed {\blue ➯ \: \sf \cfrac{2}{ \sqrt{7} + \sqrt{5} }= \sqrt{7} - \sqrt{5}}}$

$\blue{(3) ➯} \:\sf \cfrac{5}{ \sqrt{7} - \sqrt{2} } \times ( \cfrac{ \sqrt{7} + \sqrt{2} }{ \sqrt{7} + \sqrt{2} } )$

$\blue ➯ \: \sf \cfrac{5( \sqrt{7} + \sqrt{2} ) }{ \sqrt{7}^2 + { \sqrt{5} }^{2} } = \cfrac{ \cancel5( \sqrt{7} + \sqrt{2} )}{ \cancel5}$

$\underline {\boxed {\blue ➯ \: \sf \cfrac{5}{ \sqrt{7} - \sqrt{2} } = \sqrt{7} + \sqrt{2} }}$

Now, substituting and solving :-

$\blue ➯ \: \sf ( \sqrt{5} - \sqrt{2})-( \sqrt{7} - \sqrt{5} ) + ( \sqrt{7} + \sqrt{2} )$

$\blue ➯~\sf \sqrt{5} - \sqrt{2} - \sqrt{7} + \sqrt{5} + \sqrt{7} + \sqrt{2}$

$\underline {\boxed {\blue {\therefore ~\sf 2 \sqrt{5} }}}$

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@Sita05❄️