Question

if x = 1+√2 find the value of [ x - 1/x ]³

Answer 1

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Answer 2

$x = 1 + \sqrt{2} \\ given \\ \frac{1}{x} = \frac{1}{1 + \sqrt{2} } \: rationalising \\ \frac{1}{x} = \frac{1(1 - \sqrt{2)} }{(1 + \sqrt{2} )(1 - \sqrt{2}) } \\ \frac{1}{x} = \frac{1(1 - \sqrt{2)} }{ {1}^{2} - { (\sqrt[]{2} })^{2} } = \sqrt{2} - 1 \\ substracting\: both \: equation \: we \: get \\ {(x - \frac{1}{x}) }^{3} = {(1 + \sqrt{2} - ( \sqrt{2 } - 1) ) }^{3} \\ = {(1 + \sqrt{2 } - \sqrt{2} + 1) }^{3} \\ = {(2)}^{3} = 8$