Question

if x= 1+ root 2 the find the value of (x- 1/x)^3

Answer 1

$x = 1 + \sqrt{2} \\ \frac{1}{x} = \frac{1}{1 + \sqrt{2} } \\ = \frac{1}{1 + \sqrt{2} } \times \frac{1 - \sqrt{2} }{1 - \sqrt{2} } \\ = \frac{1 - \sqrt{2} }{1 - 2} \\ = \sqrt{2 } - 1 \\ x - \frac{1}{x} = 1 + \sqrt{2} - \sqrt{?} + 1 \\ 2 \\ ( {x - \frac{1}{x} })^{3} = {2}^{3} = 8$

Answer 2

see the attached document