Question

if y=1-√2,then find the value of (y-1/y)³

Answer 1

Given,

$\large \rm{}y = \orange{ 1 - \sqrt{2} }$

To find :: (y - 1/y)³

Here,

$\rm \frac{1}{y} = \frac{1}{1 - \sqrt{2} } \\ \small\boxed{ \rm rationalizing \: \: the \: \: obtained \: \: value} \\ \to \frac{1}{1 - \sqrt{2} } \times \frac{1 + \sqrt{2} }{1 + \sqrt{2} } \\ \\ \to \frac{1 + \sqrt{2} }{(1 - \sqrt{2} )(1 + \sqrt{2} )} \\ \small\boxed{ \rm using \: \: \: \rightarrow \: {\large{ \boxed{ \tt(a + b)(a - b) = {a}^{2} - {b}^{2} }}} \: \: \leftarrow \: \: identity} \\ \to \frac{1 + \sqrt{2} }{ {(1)}^{2} - {( \sqrt{2} )}^{2} } = \frac{1 + \sqrt{2} }{1 - 2} \\ \\ \to \frac{1 + \sqrt{2} }{ - 1} \: \: = \purple{ \{ - (1 + \sqrt{2} ) \}}$

Now,

$\huge \implies \rm{}y - \frac{1}{y} \\ \: \: \: \: \: \: \: \: \: \: \: \: \small \boxed{ \tt \: putting \: \: the \: \: values} \\ \large \implies \rm{}1 - \sqrt{2} - \{ - (1 + \sqrt{2} ) \} \\ \large \implies \rm{}1 - \sqrt{2} - \{ - 1 - \sqrt{2} \} \\ \large \implies \rm{}1 - \cancel{\sqrt{2} } + 1 + \cancel { \sqrt{2} } \\ \large \implies \rm2$

• Therefore,

$\implies \large \tt \bigg \{y - \frac{1}{y} \bigg\}^{3} \\ \\ \implies \tt { \big(2 \big)}^{3} \: = \red8$

Hence,

$\large\boxed{ \boxed{ \sf{ \bigg(y - \frac{1}{y} \bigg)^{3} = \: \red8} \: \: }}$