If $$\left( {x + \frac{1}{x}} \right) = 3,$$    then $$\left( {{x^2} + \frac{1}{{{x^2}}}} \right)$$   is = ?

$$\eqalign{ & {\text{Given,}} \cr & \left( {x + \frac{1}{x}} \right) = 3{\text{ }} \cr & {\text{On squaring both sides, we get}} \cr & {\left( {x + \frac{1}{x}} \right)^2} = {3^2} \cr & \Rightarrow {\left( {x + \frac{1}{x}} \right)^2} = 9 \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} + 2 = 9 \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} = 7 \cr} $$