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

Solution (By Examveda Team)

$$\eqalign{ & {\text{ }}x = \root 3 \of {2 + \sqrt 3 } \cr & {x^3} = 2 + \sqrt 3 \cr & \frac{1}{{{x^3}}} = \frac{1}{{2 + \sqrt 3 }} \times \frac{{2 - \sqrt 3 }}{{2 - \sqrt 3 }} = 2 - \sqrt 3 \cr & \therefore {x^3}{\text{ + }}\frac{1}{{{x^3}}} \cr & = 2 + \sqrt 3 + 2 - \sqrt 3 \cr & = 4 \cr} $$