Question

x = 1 - sqrt(2) find (x - 1/x) ^ 3​

Answer 1

Solution!!

x = 1 - √2

To find :- (x - 1/x)³

$\sf \left(1-\sqrt{2}-\dfrac{1}{1-\sqrt{2}}\right)^{3}$

Rationalise the denominator

$\sf = \left(1-\sqrt{2}-\left(\dfrac{1}{1-\sqrt{2}}\times \dfrac{1+\sqrt{2}}{1+\sqrt{2}}\right)\right)^{3}$

$\sf = \left(1-\sqrt{2}-\left(\dfrac{1(1+\sqrt{2})}{(1-\sqrt{2})(1+\sqrt{2})}\right)\right)^{3}$

Use the identity → (a - b)(a + b) = a² - b²

$\sf = \left(1-\sqrt{2}-\left(\dfrac{1+\sqrt{2}}{(1)^{2}-(\sqrt{2})^{2}}\right)\right)^{3}$

$\sf = \left(1-\sqrt{2}-\left(\dfrac{1+\sqrt{2}}{1-2}\right)\right)^{3}$

$\sf = \left(1-\sqrt{2}-\left(\dfrac{1+\sqrt{2}}{-1}\right)\right)^{3}$

$\sf = \left(1-\sqrt{2}-\left(-\left(1+\sqrt{2}\right)\right)\right)^{3}$

Opening the brackets and changing the sign

$\sf = \left(1-\sqrt{2}+\left(1+\sqrt{2}\right)\right)^{3}$

$\sf = \left(1-\sqrt{2}+1+\sqrt{2}\right)^{3}$

Collect the like term

$\sf = \left(1+1-\sqrt{2}+\sqrt{2}\right)^{3}$

$\sf = \left(1+1\right)^{3}$

$\sf = \left(2\right)^{3}$

$\sf = 8$