If $$x - \frac{1}{x} = 2{\text{,}}$$   then the value of the following is $${x^3} - \frac{1}{{{x^3}}}$$   = ?

Solution (By Examveda Team)

$$\eqalign{ & x - \frac{1}{x} = 2{\text{ to find }}{x^3} - \frac{1}{{{x^3}}} \cr & \Rightarrow x - \frac{1}{x} = 2 \cr & \left[ {{\text{Cubing both sides}}} \right] \cr & \Rightarrow {\left( {x - \frac{1}{x}} \right)^3} = {\left( 2 \right)^3} \cr & \Rightarrow {x^3} - \frac{1}{{{x^3}}} - 3 \times x \times \frac{1}{x}\left( {x - \frac{1}{x}} \right) = 8 \cr & \Rightarrow {x^3} - \frac{1}{{{x^3}}} - 3 \times \left( 2 \right) = 8 \cr & \Rightarrow {x^3} - \frac{1}{{{x^3}}} = 14 \cr} $$