Question

(1/√3+√2) + (1/√5-√3) - (2/√3-√2)
simplify

Answer 1

Question -

Simplify

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

Solution -

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

By rationalising the denominator we will get -

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

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

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

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

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

$= \frac{2\sqrt{3} - 2 \sqrt{2} + \sqrt{5} + \sqrt{3} - 4 \sqrt{3} - 4 \sqrt{2}} {2}$

$= \frac{3 \sqrt{3} - 4 \sqrt{3} - 2 \sqrt{2} - 4 \sqrt{2} + \sqrt{5} }{2}$

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

$= \frac{ \sqrt{5} - 6 \sqrt{2} - \sqrt{3} }{2}$