Question

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Answer 1

GIVEN :-

• (√5 - √3)/(√5 + √3)

TO RATIONALIZE :-

• (√5 - √3)/(√5 + √3)

RATIONALIZATION :-

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

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \footnotesize \displaystyle \dag \: \mathfrak{On \: Rationalizing \: we \: get,}$

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

$: \implies \displaystyle \sf \: \frac{ \big( \sqrt{5} - \sqrt{3} \big) \big( \sqrt{5} - \sqrt{3} \big)}{ \big( \sqrt{5} + \sqrt{3} \big) \big( \sqrt{5} - \sqrt{3} \big) }$

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

$: \implies \displaystyle \sf \: \frac{ \big( \sqrt{5} \big) ^{2} + \big( \sqrt{3} \big)^{2} - 2 \times \sqrt{5} \times \sqrt{3}}{ \sqrt{25} - \sqrt{9} }$

$: \implies \displaystyle \sf \: \frac{ \sqrt{25} + \sqrt{9} - 2 \sqrt{15} }{5 - 3}$

$: \implies \displaystyle \sf \: \frac{5 + 3 - 2 \sqrt{15} }{2}$

$: \implies \underline{ \boxed{\displaystyle \sf \: \bold{ \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } = \frac{8 - 2 \sqrt{15} }{2} }}}$

Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\large\boxed{\sf{Rationalize\:denominator\:\displaystyle\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}}}$

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♣ ᴀɴꜱᴡᴇʀ :

$\sf{\displaystyle\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}=\frac{8-2\sqrt{15}}{2}}$

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♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

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

$\sf{\mathrm{Multiply\:by\:the\:conjugate}\:\displaystyle\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}}$

$\sf{\displaystyle=\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}}$

Why ?

To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $\sf{ \sqrt{5}+\sqrt{3}}$

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Solve : $\sf{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}$

$\sf{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}$

$\sf{=\left(\sqrt{5}-\sqrt{3}\right)^2}$

$\sf{\text { Apply Perfect Square Formula: }(a-b)^{2}=a^{2}-2 a b+b^{2}}$

$\sf{=\left(\sqrt{5}\right)^2-2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^2}$

$\sf{=5-2\sqrt{5}\sqrt{3}+3}$

$\boxed{\sf{=8-2\sqrt{15}}}$

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Solve : $\sf{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}$

$\begin{aligned}&\sf{\text { Apply Difference of Two Squares Formula: }(a+b)(a-b)=a^{2}-b^{2}}\\\\&\sf{a=\sqrt{5}, b=\sqrt{3}}\end{aligned}$

$\sf{=\left(\sqrt{5}\right)^2-\left(\sqrt{3}\right)^2}$

$\sf{=5-3}$

$\boxed{\sf{=2}}$

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$\sf{\displaystyle\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}}$

$\large\boxed{\sf{=\displaystyle\frac{8-2\sqrt{15}}{2}}}$