Question

solve 9 question
with step by step explanation​

Answer 1

Solution :

Given $\mathrm{x = 1 - \sqrt{2}}$

Now, $\mathrm{\frac{1}{x}=\frac{1}{1-\sqrt{2}}}$

To rationalise, we multiply both the numerator and the denominator by the conjugate irrational number $\mathrm{(1+\sqrt{2})}$

$\mathrm{=\frac{1+\sqrt{2}}{(1-\sqrt{2})(1+\sqrt{2})}}$

$\mathrm{=\frac{1+\sqrt{2}}{1-2}}$

$\mathrm{=-(1+\sqrt{2})}$

So, $\mathrm{x-\frac{1}{x}}$

$\mathrm{=(1-\sqrt{2})+(1+\sqrt{2})}$

$\mathrm{=1-\sqrt{2}+1+\sqrt{2}=2}$

$\to \mathrm{x-\frac{1}{x}=2}$

Taking cubes to both sides, we get

$\mathrm{(x-\frac{1}{x})^{3}=2^{3}}$

$\to \boxed{\mathrm{(x-\frac{1}{x})^{3}=8}}$

Thus, solved!