Question

simplify

$\frac{5}{ \sqrt{3} - \sqrt{5} } + \frac{5}{ \sqrt{3} + \sqrt{5} }$
With proper explanation please !!!
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Answer 1

Solution :

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

$\tt \dashrightarrow \dfrac{5( \sqrt{3} + \sqrt{5}) + 5( \sqrt{3} - \sqrt{5} )}{( \sqrt{3} - \sqrt{5})( \sqrt{3} + \sqrt{5} ) }$

By using identity :

• $\large \underline{\boxed{\bf{(a-b)(a+b) = a^2 - b^2}}}$

$\tt \dashrightarrow \dfrac{5 \sqrt{3} + \cancel{5 \sqrt{5}} + 5 \sqrt{3} - \cancel{5 \sqrt{5}} }{ {( \sqrt{3}) }^{2} - {( \sqrt{5}) }^{2} }$

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

$\tt \dashrightarrow \dfrac{\cancel{10} \sqrt{3} }{ - \cancel{2}}$

$\tt \dashrightarrow \dfrac{5 \sqrt{3} }{ - 1}$

$\tt \dashrightarrow - 5 \sqrt{3}$

So, answer is - 5√3.

Answer 2

$\large{\underbrace{\underline{\sf{Understanding\; the\; Question}}}}$

Here in this question, concept of rationalisation is used. We are asked to simplify the given expressions. We can see that denominator of both the terms in irrational, so first we have to rationalise them by multiplying with the conjugate of their denominator to find the answer.

$\large{\tt{\longrightarrow\Bigg[\dfrac{5}{\sqrt3-\sqrt5}+\dfrac{5}{\sqrt3+\sqrt5}\Bigg]}}$

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$\large{\tt{\longrightarrow\Bigg[\dfrac{5}{\sqrt3-\sqrt5}\times\dfrac{\sqrt3+\sqrt5}{\sqrt3+\sqrt5}\Bigg]+\Bigg[\dfrac{5}{\sqrt3+\sqrt5}\times\dfrac{\sqrt3-\sqrt5}{\sqrt3-\sqrt5}\Bigg]}}$

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$\large{\tt{\longrightarrow\Bigg[\dfrac{5\Big(\sqrt3+\sqrt5\Big)}{\Big(\sqrt3-\sqrt5\Big)\Big(\sqrt3+\sqrt5\Big)}\Bigg]+\Bigg[\dfrac{5\Big(\sqrt3-\sqrt5\Big)}{\Big(\sqrt3+\sqrt5\Big)\Big(\sqrt3-\sqrt5\Big)}\Bigg]}}$

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Now apply algeberic identity:-

• [(A+B)+(A-B)=A²-B²]

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$\large{\tt{\longrightarrow\Bigg[\dfrac{5\sqrt3+5\sqrt5}{(\sqrt3)^2-(\sqrt5)^2}\Bigg]+\Bigg[\dfrac{5\sqrt3-5\sqrt5}{(\sqrt3)^2-(\sqrt5)^2}\Bigg]}}$

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Now solve the denominator

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$\large{\tt{\longrightarrow\Bigg[\dfrac{5\sqrt3+5\sqrt5}{3-5}\Bigg]+\Bigg[\dfrac{5\sqrt3-5\sqrt5}{3-5}\Bigg]}}$

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$\large{\tt{\longrightarrow\Bigg[\dfrac{5\sqrt3+5\sqrt5}{-2}\Bigg]+\Bigg[\dfrac{5\sqrt3-5\sqrt5}{-2}\Bigg]}}$

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As denominator is same of both terms, we can add numerator and take denominator same.

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$\large{\tt{\longrightarrow\Bigg[\dfrac{\Big(5\sqrt3+5\sqrt5\Big)+\Big(5\sqrt3-5\sqrt5\Big)}{-2}\Bigg]}}$

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Now simply numerator

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$\large{\tt{\longrightarrow\Bigg[\dfrac{5\sqrt3+5\sqrt5+5\sqrt3-5\sqrt5}{-2}\Bigg]}}$

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Add and subtract like terms

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$\large{\tt{\longrightarrow\Bigg[\dfrac{5\sqrt3+5\sqrt3+5\sqrt5-5\sqrt5}{-2}\Bigg]}}$

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$\large{\tt{\longrightarrow\Bigg[\dfrac{10\sqrt3}{-2}\Bigg]}}$

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Now divide -2 from both numerator and denominator

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$\large{\tt{\longrightarrow\Bigg[\dfrac{10\sqrt3\div-2}{-2\div-2}\Bigg]}}$

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$\large{\tt{\longrightarrow\Bigg[\dfrac{-5\sqrt3}{1}\Bigg]}}$

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$\large{\underline{\boxed{\sf{\longrightarrow -5\sqrt3}}}}$

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So the final answer is -5√3.