Question

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Answer 1

$Given \: \frac{ (5\sqrt{3} - 4\sqrt{2})}{(4\sqrt{3} + 3 \sqrt{2})}$

$= \frac{ (5\sqrt{3} - 4\sqrt{2})(4\sqrt{3}-3\sqrt{2})}{(4\sqrt{3} + 3 \sqrt{2})(4\sqrt{3} - 3\sqrt{2}) }$

$= \frac{ 60 -15\sqrt{6}-16\sqrt{6} +24}{(4\sqrt{3})^{2} - (3\sqrt{2})^{2}}$

$= \frac{ 84 - 31\sqrt{6}}{48 - 18}$

$= \frac{ 84 - 31\sqrt{6}}{30}$

Therefore.,

$\red {\frac{ (5\sqrt{3} - 4\sqrt{2})}{(4\sqrt{3} + 3 \sqrt{2})}}$

$\green {=\frac{ 84 - 31\sqrt{6}}{30}}$

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Answer 2

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

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

$: \star{\sf{ \: \: \frac{5 \sqrt{3} - 4 \sqrt{2} }{4 \sqrt{3} + 3 \sqrt{3} } }}$

Now, Rationalize the denominator

$: \implies{\sf{ \frac{5 \sqrt{3 } - \sqrt{2} }{4 \sqrt{3} + 3 \sqrt{2} } \times \frac{4 \sqrt{3} - 3 \sqrt{2} }{4 \sqrt{3} - 3 \sqrt{2} } }}$

$: \implies{\sf{ \frac{(5 \sqrt{3 } - \sqrt{2})(4 \sqrt{3} - 3 \sqrt{2}) }{(4 \sqrt{3} + 3 \sqrt{2} )(4 \sqrt{3} - 3 \sqrt{2} }) }}$

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

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

$: \implies{\sf{ \frac{(5 \sqrt{3 \times} \times 4 \sqrt{3} ) - (5 \sqrt{3} \times 3 \sqrt{2} ) - (4 \sqrt{2} \times 4 \sqrt{3} )(4 \sqrt{2} \times 3 \sqrt{2}) }{( {48 - 18)}}}}$

$: \implies{\sf{ \frac{(20 \times 3 ) - (15\sqrt{3 \times 2} ) - (16 \sqrt{2 \times 3 } ) + ({12 \times 2}) }{( {48 - 18)}}}}$

$: \implies{\sf{ \frac{(60 ) - (15\sqrt{6} ) - (16 \sqrt{6} ) + ({24}) }{( {48 - 18)}}}}$

$: \implies \boxed{\sf{ \frac{84 - 31 \sqrt{6} }{30} }}$

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

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

${\star \: {\sf{ \: \bigg[ \frac{1}{ {(27)}^{ \frac{ - 1}{3} } } + \frac{1}{ {(625)}^{ \frac{ - 1}{4} } } \bigg] }}}$

${\bf{\blue{\underline{Concept\:Used:}}}}$

$\star{\sf{ \: \: \: {x}^{ \frac{p}{q} } = {x}^{p \times \frac{1}{q} } }}$

$\implies{\sf{ \: \: \: {x}^{ \frac{p}{q} } = {[(x)^{p} ]}^{ \frac{1}{q} } }}$

$\star{\sf{ \: \: \: {x}^{ m} = \frac{1}{ {x}^{-m} } }}$

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

we know that ,

(3)³=27

(5)⁴=625

$: \implies{\sf{ \frac{1}{[( {3)}^{3}] ^{ \frac{ - 1}{3} } } + \frac{1}{[( {5)}^{4} ]^{ \frac{ - 1}{4} } } }}$

$: \implies{\sf{ \frac{1}{(3) ^{ \frac{ - 3}{3} } } + \frac{1}{(5) ^{ \frac{ - 4}{4} } } }}$

$: \implies{\sf{ \frac{1}{(3) ^{ - 1 } } + \frac{1}{(5) ^{ - 1} } }}$

$: \implies{\sf{ 3 + 5 }}$

$: \implies{\sf{ 8 \: Ans}}$