Question

$\huge \underline \bold \orange {Question}$

$\mathcal{PLEASE \: SOLVE \: THIS }$​

Answer 1

${\huge{\bf{\red{\underline{Question\:1:}}}}}$

${\bf{\blue{\underline{Given:}}}}$

$\dagger \: {\sf{ x = \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \: \: \: \: and \: \: y = \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } }}$

${\bf{\blue{\underline{Now:}}}}$

Take,

$: \implies{\sf{x = \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } }}$

Rationalize,

$: \implies{\sf{x = \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } }}$

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

$\bigstar \: \boxed{\sf{{x}^{2} - {y}^{2} = (x - y)(x + y)}}$

$\bigstar \: \boxed{\sf{ \purple{{(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy}}}$

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

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

$: \implies{\sf{ \frac{8 + 2 \sqrt{3} \sqrt{5} }{2} }}$

$: \implies{\sf{ \frac{ \cancel2(4 + \sqrt{3} \sqrt{5}) }{ \cancel2} }}$

$: \implies{\sf{ 4 + \sqrt{3 \times 5} }}$

$: \implies \boxed{\sf{ x = 4 + \sqrt{15}}}$

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Now,

$: \implies{\sf{ y = \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } }}$

Rationalize,

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

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

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

$: \implies{\sf{ \frac{8 - 2\sqrt{15} }{2} }}$

$: \implies{\sf{ \frac{ \cancel2(4 - \sqrt{3} \sqrt{5}) }{ \cancel2} }}$

$: \implies{\sf{ 4 - \sqrt{3 \times 5} }}$

$: \implies \boxed{\sf{ x = 4 - \sqrt{15}}}$

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$: \implies{\sf{ {x}^{2} + {y}^{2} }}$

$: \implies{\sf{ {(4 + \sqrt{15}) }^{2} + {(4 - \sqrt{15} })^{2} }}$

$: \implies{\sf{ (4 + \sqrt{15} + 4 - \sqrt{15} ) ^{2} - 2 \times[ (4 + \sqrt{15})(4 - \sqrt{15} ) ] }}$

$: \implies{\sf{ (4 + 4 ) ^{2} - 2 [\times (4 + \sqrt{15})(4 - \sqrt{15} ) ]}}$

$: \implies{\sf{ (8) ^{2} - 2 \times ({4}^{2} - ( \sqrt{15} ) ^{2} )}}$

$: \implies{\sf{ 64 - 2 \times( 16 - 15)}}$

$: \implies{\sf{ 64 - 2}}$

$: \implies \boxed{\sf{ {x}^{2} + {y}^{2} = 62}}$

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${\huge{\bf{\red{\underline{Question\:2:}}}}}$

${\bf{\blue{\underline{Given:}}}}$

$\star \: \: {\sf{ \green{ \bigg( \frac{81}{16} \bigg)^{ \frac{ - 3}{4} } \times \bigg \{ \bigg( \frac{25}{9} \bigg) ^{ \frac{ - 3}{2} } \div \bigg( \frac{5}{2} \bigg) ^{ - 3} \bigg\} =1}} }$

Take L.H.S

$: \implies \: \: {\sf{ \bigg( \frac{( {3)}^{4} }{( {2)}^{4} } \bigg)^{ \frac{ - 3}{4} } \times \bigg \{ \bigg( \frac{( {5)}^{2} }{( {3)}^{2} } \bigg) ^{ \frac{ - 3}{2} } \div \bigg( \frac{5}{2} \bigg) ^{ - 3} \bigg\} }}$

$: \implies \: \: {\sf{ \bigg( \frac{( {3)} }{( {2)}} \bigg)^{ 4 \times \frac{ - 3}{4} } \times \bigg \{ \bigg( \frac{( {5)} }{( {3)} } \bigg) ^{ 2 \times \frac{ - 3}{2} } \div \bigg( \frac{5}{2} \bigg) ^{ - 3} \bigg\} }}$

$: \implies \: \: {\sf{ \bigg( \frac{( {3)} }{( {2)}} \bigg)^{ - 3 } \times \bigg \{ \bigg( \frac{( {5)} }{( {3)} } \bigg) ^{ - 3 } \div \bigg( \frac{5}{2} \bigg) ^{ - 3} \bigg\} }}$

$: \implies \: \: {\sf{ \bigg( \frac{ {2} }{{3}} \bigg)^{ 3 } \times \bigg \{ \bigg( \frac{{3} }{ {5} } \bigg) ^{ 3 } \div \bigg( \frac{2}{5} \bigg) ^{ 3} \bigg\} }}$

$: \implies \: \: {\sf{ \bigg( \frac{ {2} }{{3}} \bigg)^{ 3 } \times \bigg \{ \bigg( \frac{{3} }{ {5} } \bigg) ^{ 3 } \times \bigg( \frac{5}{2} \bigg) ^{ 3} \bigg\} }}$

$: \implies \: \: {\sf{ \bigg( \frac{ {2} }{{3}} \bigg)^{ 3 } \times \bigg ( \frac{{3} }{ {5} } \bigg) ^{ 3 } \times \bigg( \frac{5}{2} \bigg) ^{ 3} }}$

$: \implies \: \: {\sf{ \bigg( \frac{ {2} }{{3}} \times \frac{{3} }{ {5} } \times \frac{5}{2} \bigg) ^{ 3} }}$

$: \implies{\sf{ (2) ^{3} \times ( {2)}^{ - 3} \times {(3)}^{3} \times ( {3)}^{ - 3} \times ( {5)}^{3} \times ( {5)}^{ - 3} }}$

$: \implies{\sf{ ( {2)}^{ 3- 3} \times ( {3)}^{ 3- 3} \times ( {5)}^{3 - 3} }}$

$: \implies{\sf{ ( {2)}^{ 0} \times ( {3)}^{ 0} \times ( {5)}^{0} }}$

$: \implies{\sf{ 1}}$

Hence, L.HS=R.HS

Answer 2

$\mathfrak{\huge{\red{\underline{\underline{Answer}}}}}$

$\small\sf\blue{\underline{Both \: answers \: are \: there \: in \: the}{ \underline{ \: attachment \: above \: }}}$