Question

show that
1/(2-√3) - 1/(√3-√2)+1/(√2-1)​

Answer 1

Given

$\sf \to \: \dfrac{1}{(2 - \sqrt{3}) } - \dfrac{1}{( \sqrt{3} - \sqrt{2} )} + \dfrac{1}{( \sqrt{2} - 1)} = 3$

Now Take

$\sf \to \: \dfrac{1}{(2 - \sqrt{3}) } - \dfrac{1}{( \sqrt{3} - \sqrt{2} )} + \dfrac{1}{( \sqrt{2} - 1)}$

And Use Rationalization Method

$\sf \to \: \dfrac{1}{(2 - \sqrt{3}) } \times \dfrac{(2 + \sqrt{3} )}{(2 + \sqrt{3}) } - \dfrac{1}{( \sqrt{3} - \sqrt{2} )} \times \dfrac{( \sqrt{3} + \sqrt{2})}{( \sqrt{3} + \sqrt{2})} + \dfrac{1}{( \sqrt{2} - 1)} \times \dfrac{ \sqrt{2} + 1 }{ \sqrt{2 + 1} }$

We Use this identities

$\sf \to(a - b)(a + b) = {a}^{2} - {b}^{2}$

We get

$\sf \to \: \dfrac{(2 + \sqrt{3}) }{(2 - \sqrt{3})(2 + \sqrt{3})} - \dfrac{ (\sqrt{3} + \sqrt{2}) }{( \sqrt{3} - \sqrt{2})( \sqrt{3} + \sqrt{2} )} + \dfrac{( \sqrt{2} - 1) }{( \sqrt{2} - 1)( \sqrt{2} + 1)}$

$\sf \to \: \dfrac{(2 + \sqrt{3}) }{(2) {}^{2} - (\sqrt{3})^{2}}- \dfrac{ (\sqrt{3} + \sqrt{2}) }{( \sqrt{3} ) {}^{2} - (\sqrt{2}) {}^{2} } + \dfrac{( \sqrt{2} - 1) }{( \sqrt{2})^{2} - 1)}$

$\sf \to \: \dfrac{2 + \sqrt{3} }{4 - 3} - \dfrac{ \sqrt{3} + \sqrt{2} }{3 - 2} + \dfrac{ \sqrt{2} + 1 }{2 - 1}$

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

$\sf \to \: 2 + \sqrt{3} - \sqrt{3} - \sqrt{2} + \sqrt{2} + 1$

$\sf \to \: 2 + 1$

$\sf \to \: 3$

Hence proved