Question

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Answer 1

Given:-

• $\rm{\dfrac{1}{2 + \sqrt{3}} + \dfrac{2}{\sqrt{5} - \sqrt{3}} + \dfrac{1}{ 2 - \sqrt{5}}}$

To Find:-

• The Value after Simplifying.

Method Used:-

• Rationalisation.

Now,

Let's take the first number.

$\rm{ \dfrac{1}{2 + \sqrt{3}}\times{\dfrac{2 - \sqrt{3}}{ 2 - \sqrt{3}}}}$

$\rm{ \dfrac{ 2 - \sqrt{3}} { 1} }$.....1

Now, Take second number.

$\rm{ \dfrac{2}{\sqrt{5} - \sqrt{3}}\times{\dfrac{\sqrt{5} +\sqrt{3}}{ \sqrt{5} +\sqrt{3}}}}$

$\rm{\dfrac{2(\sqrt{5} + \sqrt{3})}{2}}$

$\rm{\dfrac{\cancel{2}(\sqrt{5} + \sqrt{3})}{\cancel{2}}}$

$\rm{\sqrt{5} + \sqrt{3}}$......2

Now, Take Third Number.

$\rm{ \dfrac{1}{2 - \sqrt{5}}\times{\dfrac{2+\sqrt{5}}{ 2 +\sqrt{5}}}}$

$\rm{\dfrac{ 2 + \sqrt{5}} {1}}$

$\rm{ 2 + \sqrt{5}}$.....3

Now adding 1, 2 and 3

$\rm{ 2 - \sqrt{3}}$ + $\rm{\sqrt{5} + \sqrt{3}}$ + $\rm{ 2 + \sqrt{5}}$

$\rm{ 2 - \cancel{\sqrt{3}}}$ + $\rm{\sqrt{5} + \cancel{\sqrt{3}}}$ + $\rm{ 2 + \sqrt{5}}$

$\rm{ 4 + 2\sqrt{5}}$.

Hence, The Simplified value is 4 + 2√5.

Answer 2

$\huge{ \bf{ \red{\mid{ \overline{ \underline{ANSWER}}} \mid}}}$

$\huge \: \tt{4 + 2 \sqrt{5} }$

$\bf {EXPLANATION \: - }$

$\bf{ \pink{GIVEN :-}} \\ \\ \bf{ \: \frac{1}{2 + \sqrt{3} } + \frac{2}{ \sqrt{5} - \sqrt{3} } + \frac{1}{2 - \sqrt{5}} }$

let's Rationalise each one -

$\sf \implies \: \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \\ \sf \implies \: 2 - \sqrt{3} ......(i)$

$\sf \implies \: \frac{2}{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} } { \sqrt{5} + \sqrt{3} } \\ \\ \sf \implies \: \frac{ \cancel{2}( \sqrt{5} + \sqrt{3} ) }{ \cancel{2} } \: = \sqrt{5} + \sqrt{3} ..(ii)$

$\sf \implies \frac{1}{2 - \sqrt{5} } \times \frac{2 + \sqrt{5} }{2 + \sqrt{5} } \\ \\ \sf \implies \: 2 + \sqrt{5} ...(iii)$

Add equations ( i ) , ( ii ) & ( iii )

→ 2 - √3 + √5 + √ 3 + 2 + √ 5

→ 4 + 2√5

$\mathfrak {\implies \: hence \: the \: answer \: is \: } \\ \\ \bf \implies \: \mathfrak{4 + 2 \sqrt{5} }$