Question

If $x=1-\sqrt{2},$ find the value of ${(x-\frac{1}{x})}^3.$

• A. $7$
• B. $8$
• C. $5$
• D. $6$

Answer 1

The correct option is B $8$

$x=1-\sqrt{2}$

$\frac{1}{x}=\frac{1}{1-\sqrt{2}}$

We rationalise the denominator, we get

$\frac{1}{x}=\frac{1}{1-\sqrt{2}}\times \frac{1+\sqrt{2}}{1+\sqrt{2}}$

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

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

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

$\therefore {(x-\frac{1}{x})}^3=(2{)}^3=8$