Question

Find the value of (a-1/a) if a=1-root2

Answer 1

given :- a = 1 - √2

$\sf \therefore \: \frac{1}{a} = \frac{1}{1 - \sqrt{2} } \\ \\ \sf rationalizing \\ \\ \sf = \frac{1}{1 - \sqrt{2} } \times \frac{1 + \sqrt{2} }{1 + \sqrt{2} } \\ \\ \sf = \frac{1 + \sqrt{2} }{(1 - \sqrt{2})(1 + \sqrt{2} ) } \\ \\ \sf = \frac{1}{( {1})^{2} - ( { \sqrt{2}) }^{2} } \\ \\ \sf = \frac{1 + \sqrt{2} }{1 - 2} \\ \\ \sf = \frac{1 + \sqrt{2} }{ - 1} \\ \\ \sf = - (1 + \sqrt{2} ) = - 1 - \sqrt{2}$

$\sf hence \: value \: of \: x - \frac{1}{x} : - \\ \\ \sf = 1 - \sqrt{2} - ( - 1 - \sqrt{2} ) \\ \sf = 1 - \cancel{ \sqrt{2} } + 1 + \cancel{\sqrt{2} } \\ \sf = 1 + 1 \\ \sf = \boxed2$