Question

X=2+√3
Find (x-1/x)^3

Answer 1

$x = 2 + \sqrt{3} \\ \frac{1}{x} = \frac{1}{2 + \sqrt{3} } = \frac{2 - \sqrt{3} }{(2 + \sqrt{3})(2 - \sqrt{3} ) } \\ = \frac{2 - \sqrt{3} }{4 - 3} = 2 - \sqrt{3} \\ therefore \\ x - \frac{1}{x} = (2 + \sqrt{3} ) - (2 - \sqrt{3} ) \\ = 2 \sqrt{3} \\ therefore \\ {(x - \frac{1}{x}) }^{3} = {(2 \sqrt{3} )}^{3} = 24 \sqrt{3}$