Question

Rationalise the denominator of the following and show the steps
$\frac{5\sqrt{3} +3\sqrt{5} }{ 5\sqrt{3} -3\sqrt{5} }$


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Answer 1

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Answer 2

$\large {\underline {\underline {\sf {Question}}}}$

Rationalise the denominator od the following:-

$\implies \dfrac{5\sqrt{3} + 3\sqrt{5}}{5\sqrt{3} - 3\sqrt{5}}$

$\large {\underline {\underline {\sf {\pink {How\: to \: solve?}}}}}$

• To rationalize a denominator we have to multiply the denominator with a radical opposite the sing given.
• then we have to solve further by identifying the algebric identities.
• finally we are with the answer.
• So, lets start

$\large {\underline {\underline {\sf {\blue {Solution}}}}}$

$\implies \tt \dfrac {5\sqrt{3} + 3\sqrt{5}} {5\sqrt{3} - 3\sqrt{5}}$

$\implies \tt multiply \: denominator \: by\: radical$

$\implies \dfrac{5\sqrt{3} + 3\sqrt{5}} {5\sqrt{3} - 3\sqrt{5}} \times \dfrac{5\sqrt{3} + 3\sqrt{5}} {5\sqrt{3} + 3\sqrt{5}}$

$\implies \tt \dfrac{5\sqrt{3} + 3\sqrt{5}²} {5\sqrt{3}² - 3\sqrt{5}²}$

$\implies \tt by \: using\: algebric\: identity \\ \tt (a +b)² = a² +b² +2ab$

$\implies\dfrac{5\sqrt{3}² + 2 \times 5\sqrt{3} \times 3\sqrt{5} + 3\sqrt{5}² } {75 - 45}$

$\implies \tt \dfrac{75 + 30\sqrt{15} + 45}{30}$

$\implies \tt \dfrac{15 ( 5 + 2\sqrt{15} +3)}{30}$

$\implies \tt \dfrac{ 8 + 2\sqrt{15}}{2}$

$\implies \tt 4 + \sqrt{15}$

$\large {\boxed {\boxed {\tt {\green {Answer = 4 + \sqrt{15}}}}}}$

Hence, we are done with the problem.