Step 1 :\( \begin{array}{c} a^{8}+\frac{1}{a^{8}}+2=2+\sqrt{2} \end{array} \)
Step 2 :\( \begin{array}{c} a^{8}+\frac{1}{a^{8}}=\sqrt{2} \end{array} \)
Step 3 :\( \begin{array}{c} \left(a^{8}+\frac{1}{a^{8}}\right)^{2}=2 \Rightarrow a^{16}+2+\frac{1}{a^{16}}=2 \end{array} \)