Question

Simplify 1/√3+√2-√5​

Answer 1

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

_________ [GIVEN EQUATION]

• We have to simply it.

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=> $\dfrac{1}{ \sqrt{3} \: + \: \sqrt{2} \: - \: \sqrt{5} }$

• Rationalize it.

=> $\dfrac{1}{ (\sqrt{3} \: + \: \sqrt{2}) \: - \: \sqrt{5} }$ × $\dfrac{ (\sqrt{3} \: + \: \sqrt{2} ) \: + \sqrt{5} }{ (\sqrt{3} \: + \: \sqrt{2}) \: + \: \sqrt{5} }$

=> $\dfrac{( \sqrt{3} \: + \: \sqrt{2}) \: + \: \sqrt{5} }{ {( \sqrt{3} \: + \: \sqrt{2} ) }^{2} \: + \: {( \sqrt{5} }^{2} )}$

(a + b)² = a² + b² + 2ab

=> $\dfrac{( \sqrt{3} \: + \: \sqrt{2}) \: + \: \sqrt{5} }{ 3 \: + \: 2\: + \: 2(3)(2) \: + \: {5}}$

=> $\dfrac{( \sqrt{3} \: + \: \sqrt{2}) \: + \: \sqrt{5} }{ 3 \: + \: 2\: + \: 2(\sqrt{2)}\: (\sqrt{3)}\: - \: {5}}$

=> $\dfrac{( \sqrt{3} \: + \: \sqrt{2}) \: + \: \sqrt{5} }{ 6 \:+\:2\sqrt{6}\: - \: {5}}$

=> $\dfrac{( \sqrt{3} \: + \: \sqrt{2}) \: + \: \sqrt{5} }{2\sqrt{6}}$

=> $\dfrac{( \sqrt{3} \: + \: \sqrt{2}) \: + \: \sqrt{5} }{2\sqrt{6}}$ × $\dfrac{\sqrt{6}}{\sqrt{6}}$

=> √6 × $\dfrac{( \sqrt{3} \: + \: \sqrt{2}) \: +\: \sqrt{5} }{12}$

=> $\dfrac{ \sqrt{18} \: + \: \sqrt{12} \: +\: \sqrt{30} }{12}$

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Answer 2

Answer:

$= > \frac{ \sqrt{2} + \sqrt{3} + \sqrt{30} }{2}$

Step-by-step explanation:

We have ,

$= > \frac{1}{ \sqrt{3} + \sqrt{2} - \sqrt{5} }$

$= > \frac{1}{ \sqrt{3} + \sqrt{2} - \sqrt{5} } \times \frac{( \sqrt{3 + 2 } ) + \sqrt{5} }{( \sqrt{3} + \sqrt{2} ) + \sqrt{5} }$

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

$= > \frac{ \sqrt{3} + \sqrt{2} + \sqrt{5} }{(5 + 2 \sqrt{6} ) - 5 }$

$= > \frac{ \sqrt{3} + \sqrt{2} + \sqrt{5} }{ 2 \sqrt{6} } \times \frac{ \sqrt{6} }{ \sqrt{6} }$

$= > \frac{ \sqrt{18} + \sqrt{12} + \sqrt{30} }{2 \times 6}$

$= > \frac{3 \sqrt{2} + 2 \sqrt{3} + \sqrt{30} }{12}$

$= > \frac{ \sqrt{2} + \sqrt{3} + \sqrt{30} }{2}$