Question

1. Ifx=1- root2 , find the value of (x-1/x)

Answer 1

Question :

If x = 1 - √2 , find the value of (x - 1/x)³

Solution :

We know the value of x then in 1/x we will get

$\rarr x = \frac{1}{1 - \sqrt{2} }$

$\looparrowright \frac{1}{x} = \frac{1(1 + \sqrt{2}) }{(1 - \sqrt{2})(1 + \sqrt{2}) }$

$\looparrowright \frac{1}{x} = \frac{1 + \sqrt{2} }{ {(1)}^{2} - {( \sqrt{2}) }^{2} }$

$\looparrowright \frac{1}{x} = \frac{1 + \sqrt{2} }{1 - 2}$

$\looparrowright \frac{1}{x} = - 1 - \sqrt{2}$

Now simplifying x - 1/x

$\leadsto x - \frac{1}{x}$

Putting the values we get

$\leadsto 1 - \sqrt{2} - ( - 1 - \sqrt{2} )$

$\leadsto 1 - \sqrt{2} + 1 + \sqrt{2}$

$\leadsto1 + 1 + \cancel{ \sqrt{2} } - \cancel{ \sqrt{2} }$

$\leadsto2$

So by cubing x - 1/x we get

$\rarr {(x - \frac{1}{x} )}^{3}$

$\rarr {(2)}^{3}$

$\rarr \boxed{8}$

So the value is 8