Question

I need help
if x = 1-√2, then find the value of x³+1/x³

Answer 1

$x = 1 - \sqrt{2} \\ \frac{1}{x} = \frac{1}{1 - \sqrt{2} } = \frac{1 + \sqrt{2} }{(1 - \sqrt{2} )(1 + \sqrt{2} )} \\ = \frac{1 + \sqrt{2} }{1 - 2} = - 1 - \sqrt{2} \\ therefore \: \: \\ x + \frac{1}{x} = (1 - \sqrt{2} ) + ( - 1 - \sqrt{2} ) = - 2 \sqrt{2} \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } = {(x + \frac{1}{x} )}^{3} - 3(x + \frac{1}{x} ) \\ = {( - 2 \sqrt{2} )}^{3} - 3 \times - 2 \sqrt{2} \\ = - 16 \sqrt{2} + 6 \sqrt{2} = - 10 \sqrt{2}$