If $$\left( {a + \frac{1}{a}} \right) = 6,$$    then $$\left( {{a^4} + \frac{1}{{{a^4}}}} \right)$$   = ?

$$\eqalign{ & \left( {a + \frac{1}{a}} \right) = 6 \cr & \Rightarrow {\left( {a + \frac{1}{a}} \right)^2} = {6^2} \cr & \,\,\,\,\,\,\,\, = 36 \cr & \Rightarrow {a^2} + \frac{1}{{{a^2}}} + 2 = 36 \cr & \Rightarrow \left( {{a^2} + \frac{1}{{{a^2}}}} \right) = 34 \cr & \Rightarrow {\left( {{a^2} + \frac{1}{{{a^2}}}} \right)^2} = {\left( {34} \right)^2} \cr & \Rightarrow {a^4} + \frac{1}{{{a^4}}} + 2 = 1156 \cr & \Rightarrow \left( {{a^4} + \frac{1}{{{a^4}}}} \right) = 1154 \cr} $$