If $$a = 2 + \sqrt 3 {\text{,}}$$   then the value of $$\left( {{a^2} + \frac{1}{{{a^2}}}} \right) = \,?$$

$$\eqalign{ & a = 2 + \sqrt 3 \cr & \Rightarrow {a^2} = {\left( {2 + \sqrt 3 } \right)^2} \cr & \Rightarrow {a^2} = 4 + 3 + 4\sqrt 3 \cr & \Rightarrow {a^2} = 7 + 4\sqrt 3 \cr & \frac{1}{{{a^2}}} = \frac{1}{{7 + 4\sqrt 3 }} \cr & \Rightarrow \frac{1}{{{a^2}}} = \frac{{7 - 4\sqrt 3 }}{{\left( {7 + 4\sqrt 3 } \right)\left( {7 - 4\sqrt 3 } \right)}} \cr & \Rightarrow \frac{1}{{{a^2}}} = \frac{{7 - 4\sqrt 3 }}{1} \cr & \Rightarrow \frac{1}{{{a^2}}} = 7 - 4\sqrt 3 \cr & \therefore {a^2} + \frac{1}{{{a^2}}} \cr & = 7 + 4\sqrt 3 + 7 - 4\sqrt 3 \cr & = 14 \cr} $$