Question

if x=1-√2,find the value of [x-1/x]³​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{(x-\frac{1}{x})^{3}=8}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies x = 1 - \sqrt{2} \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies( x- \frac{1}{x} )^{3} =?$

• According to given question :

$\bold{as \: we \: know \: that} \\ \tt: \implies {(x - \frac{1}{x} )}^{3} \\ \\ \tt: \implies {(1 - \sqrt{2} - \frac{1}{1 - \sqrt{2} })}^{3} \\ \\ \tt: \implies {(1 - \sqrt{2} - \frac{1}{1 - \sqrt{2} } \times \frac{1 + \sqrt{2} }{1 + \sqrt{2} } ) }^{3} \\ \\ \tt: \implies {(1 - \sqrt{2} - \frac{1 + \sqrt{2} }{ {1}^{2} - { (\sqrt{2})}^{2} } )}^{3} \\ \\ \tt: \implies {(1 - \sqrt{2} - \frac{1 - \sqrt{2} }{1 - 2} )}^{3} \\ \\ \tt: \implies ( { 1 - \sqrt{2} - \frac{1 - \sqrt{2} }{ - 1} )}^{3} \\ \\ \tt: \implies ( {1 - \sqrt{2} + 1 + \sqrt{2} )}^{3} \\ \\ \tt: \implies (1 + 1)^{3} \\ \\ \tt: \implies {2}^{3} \\ \\ \green{\tt: \implies 8} \\ \\ \green{\tt \therefore {(x - \frac{1}{x} )}^{3} = 8}$

Answer 2

ANSWER:

x = 1 - √2

(x - 1/x)³

Substituting the value of x

$\to \tt{{\bigg[1 - \sqrt{2} -\frac{1 }{1 - \sqrt{2}}}\bigg]^3}\\ \\ \\ \to \tt{\bigg[{\frac{(1 - \sqrt{2})^2 - 1 }{1 - \sqrt{2}}} \bigg]^3}$

Simplifying using

★ (a - b)² = a² - 2ab + b²

$\to \tt{\bigg[\frac{1^2 - 2(1)(\sqrt{2}) + (\sqrt{2})^2 - 1}{1 - \sqrt{2}}\bigg]^3}$

→[ 2 - 2√2/1 - √2]³

→ [2(1 - √2)/1 - √2]³

→ 2³

→ 8

Hence , (x - 1/x)³ = 8