1.

Simplify 1/√3+√2-√5​

Answer»

\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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