If $$x + \frac{1}{x} = \sqrt 3 {\text{,}}$$   then find the value of $${x^3} + \frac{1}{{{x^3}}}$$   = ?

Explanation:-

$$\eqalign{ & x + \frac{1}{x} = \sqrt 3 \cr & {\text{Cubing both side}} \cr & \Rightarrow {x^3} + \frac{1}{{{x^3}}} + 3.x.\frac{1}{x}\left( {x + \frac{1}{x}} \right) = 3\sqrt 3 \cr & \Rightarrow {x^3} + \frac{1}{{{x^3}}} + 3\left( {\sqrt 3 } \right) = 3\sqrt 3 \cr & \Rightarrow {x^3} + \frac{1}{{{x^3}}} = 3\sqrt 3 - 3\sqrt 3 \cr & \Rightarrow {x^3} + \frac{1}{{{x^3}}} = 0 \cr}$$