Question

Solve this :
$\Large \frac{2}{ \sqrt{7} + \sqrt{5} } + \frac{7}{ \sqrt{12} - \sqrt{5} } - \frac{5}{ \sqrt{12} + \sqrt{7} }$
​

Answer 1

AnswEr :

$\longrightarrow \sf \small\dfrac{2}{ \sqrt{7} + \sqrt{5} } + \dfrac{7}{ \sqrt{12} - \sqrt{5} } - \dfrac{5}{ \sqrt{12} + \sqrt{7} }$

• Rationalizing Denominators

$\longrightarrow \sf \small \bigg(\dfrac{2}{ \sqrt{7} + \sqrt{5} } \times \dfrac{\sqrt{7} - \sqrt{5}}{ \sqrt{7} - \sqrt{5} } \bigg)+ \bigg(\dfrac{7}{ \sqrt{12} - \sqrt{5} }\times \dfrac{\sqrt{12} + \sqrt{5}}{ \sqrt{12} + \sqrt{5}}\bigg) -\bigg(\dfrac{5}{ \sqrt{12} + \sqrt{7} }\times \dfrac{\sqrt{12} - \sqrt{7}}{ \sqrt{12} - \sqrt{7}}\bigg)$

$\longrightarrow \small \sf \dfrac{2( \sqrt{7} - \sqrt{5} ) }{ ({ \sqrt{7}) }^{2} - { (\sqrt{5} )}^{2} } + \dfrac{7( \sqrt{12} + \sqrt{5} ) }{ ({ \sqrt{12}) }^{2} - { (\sqrt{5} )}^{2} } - \dfrac{5( \sqrt{12} - \sqrt{7} ) }{ ({ \sqrt{12}) }^{2} - { (\sqrt{7} )}^{2} }$

$\longrightarrow \small \sf \dfrac{ \cancel2( \sqrt{7} - \sqrt{5} ) }{ \cancel2} + \dfrac{ \cancel7( \sqrt{12} + \sqrt{5} ) }{ \cancel7 } - \dfrac{ \cancel5( \sqrt{12} - \sqrt{7} ) }{ \cancel5}$

$\longrightarrow \sf( \sqrt{7} - \sqrt{5} ) + ( \sqrt{12} + \sqrt{5} ) - ( \sqrt{12} - \sqrt{7} )$

$\longrightarrow \sf\sqrt{7} - \cancel{\sqrt{5}} + \cancel{\sqrt{12}} + \cancel{\sqrt{5}} - \cancel{\sqrt{12}} + \sqrt{7}$

$\longrightarrow \sf \sqrt{7} + \sqrt{7}$

$\longrightarrow \large \sf 2 \sqrt{7}$

⠀

$\therefore$ Value will be Equal to 2√7.

Answer 2

SOLUTION

$\frac{2}{ \sqrt{7} + \sqrt{5} } + \frac{7}{ \sqrt{12} - \sqrt{5} } - \frac{5}{ \sqrt{12} + \sqrt{7} } \\ \\ we \: can \: write \: \\ 2 \: as \: {( \sqrt{7} })^{2} - {( \sqrt{5} )}^{2} \\ 7 \: as \: {( \sqrt{12}) }^{2} - {( \sqrt{5}) }^{2} \\ 5 \: as \: {( \sqrt{12} )}^{2} - {( \sqrt{7} )}^{2} \\ \\ now \: rewrite \: each \: term \: of \: the \: question \\ \\ \frac{ ({ \sqrt{7} )}^{2} - {( \sqrt{5} )}^{2} }{ \sqrt{7} + \sqrt{5} } + \frac{ {( \sqrt{12} )}^{2} - {( \sqrt{5} )}^{2} }{ \sqrt{12} - \sqrt{5} } - \frac{ {( \sqrt{12}) }^{2} - {( \sqrt{7} )}^{2} }{ \sqrt{12} + \sqrt{7} } \\ \\ we \: also \: know \: that \\ {a}^{2} - {b}^{2} = (a + b)(a - b) \\ \\ so \: from \: this \: we \: can \: write \: \\ \\ \frac{ ({ \sqrt{7}) }^{2} - {( \sqrt{5} )}^{2} }{ \sqrt{7} + \sqrt{5} } = \sqrt{7} - \sqrt{5} \\ \\ \frac{ ({ \sqrt{12}) }^{2} - {( \sqrt{5} )}^{2} }{ \sqrt{12} - \sqrt{5} } = \sqrt{12} + \sqrt{5} \\ \\ \frac{ {( \sqrt{12}) }^{2} - {( \sqrt{7} )}^{2} }{ \sqrt{12} + \sqrt{7} } = \sqrt{12} - \sqrt{7} \\ \\ now \: the \: question \: turns \: to \\ \\ \sqrt{7} - \sqrt{5} + \sqrt{12} + \sqrt{5} - ( \sqrt{12} </u></strong><strong><u>-</u></strong><strong><u> \sqrt{7} ) \\ \\ answer \: is \: </u></strong><strong><u>2</u></strong><strong><u>\</u></strong><strong><u>s</u></strong><strong><u>q</u></strong><strong><u>r</u></strong><strong><u>t</u></strong><strong><u>{</u></strong><strong><u>7</u></strong><strong><u>}</u></strong><strong><u>....acc \: to \: me$