Question

please answer it i am not getting it....​

Answer 1

EXPLANATION.

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

As we know that,

Solve this equation one by one, we get.

Rationalizes the equation, we get.

$\implies \dfrac{1}{\sqrt{3} + \sqrt{2} }$

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

$\implies \dfrac{\sqrt{3} - \sqrt{2} }{[(\sqrt{3})^{2} - (\sqrt{2} )^{2} ] }$

$\implies \dfrac{\sqrt{3}- \sqrt{2} }{[3 - 2]} = \sqrt{3} - \sqrt{2} . - - - - - (1).$

$\implies \dfrac{2}{\sqrt{5}- \sqrt{3} }$

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

$\implies \dfrac{2[\sqrt{5}+ \sqrt{3} ]}{[(\sqrt{5})^{2} - (\sqrt{3} )^{2} }$

$\implies \dfrac{2[\sqrt{5} + \sqrt{3} ]}{5 - 3 } = \dfrac{2[\sqrt{5}+ \sqrt{3} ] }{2}$

$\implies \sqrt{5} + \sqrt{3} . - - - - - (2).$

$\implies \dfrac{3}{\sqrt{2}- \sqrt{5} }$

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

$\implies \dfrac{3[\sqrt{2} + \sqrt{5}] }{[(\sqrt{2})^{2} - (\sqrt{5} )^{2}] }$

$\implies \dfrac{3[\sqrt{2} + \sqrt{5}] }{2 - 5 } = \dfrac{3[\sqrt{2}+ \sqrt{5} ]}{-3}$

$\implies - [\sqrt{2} + \sqrt{5} ]. - - - - - (3).$

Write equation in original form, we get.

$\implies [\sqrt{3} - \sqrt{2} ] - [\sqrt{5} + \sqrt{3} ] - [- (\sqrt{2} + \sqrt{5} )].$

$\implies \sqrt{3} - \sqrt{2} - \sqrt{5} - \sqrt{3} + \sqrt{2} + \sqrt{5} .$

$\implies 0$

Answer 2

$\large\bf\underline\red{Answer \: :-}$

Here, we three fraction we solve all fraction one by one, first rationalize the all fraction.

1) 1/√3+√2

$\sf\implies{ \frac{1}{ \sqrt{3} + \sqrt{2} } \times \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} - \sqrt{2} } } \\ \\ \sf\implies{ \frac{ \sqrt{3} - \sqrt{2} }{( \sqrt{3} ) {}^{2} - (\sqrt{2}) {}^{2} } } \\ \\ \sf\implies{ \sqrt{3} - \sqrt{2} \: \: \: \: \: ( \because3 - 2 = 1)} \\ \\ \sf\implies \pink{ \sqrt{3} - \sqrt{2} \: \: \: \: \: \: \: \: \: ...(1) }$

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2) 2/√5-√3

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

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3) 3/√2-√5

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

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Now,

$\small\sf\dashrightarrow{ \sqrt{3} - \sqrt{2} - ( \sqrt{5} + \sqrt{3}) - ( - \sqrt{2} - \sqrt{5} ) } \\ \\ \small\sf\dashrightarrow{ \sqrt{3} - \sqrt{2} - \sqrt{5} - \sqrt{3} + \sqrt{2} + \sqrt{5} } \: \: \: \: \: \: \: \: \: \\ \\ \small\sf\dashrightarrow{ \sqrt{3} \: \: \cancel{ - \sqrt{2} } - \sqrt{5} - \sqrt{3} \: \: \cancel{ + \sqrt{2} } + \sqrt{5} } \: \: \: \: \: \: \: \: \\ \\ \small\sf\dashrightarrow{ \sqrt{ 3} \: \: \cancel{ - \sqrt{5} } - \sqrt{3} \: \: \cancel{ + \sqrt{5} }} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \small\sf\dashrightarrow{ \sqrt{3} - \sqrt{3} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \small\sf\dashrightarrow{0} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

0 is the answer.

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