Solution
$\mathrm{x}^{2}-\sqrt{2} \mathrm{x}+2=0 \Rightarrow \mathrm{x}=\frac{\sqrt{2} \pm \sqrt{6} \mathrm{i}}{2} \Rightarrow \alpha=\frac{\sqrt{2}+\sqrt{6} \mathrm{i}}{2}=\sqrt{2} \mathrm{e}^{\frac{\mathrm{i} \pi}{3}}$
\[
\begin{aligned}
& \beta=\sqrt{2} \mathrm{e}^{-\frac{\mathrm{i} \pi}{3}} \Rightarrow \alpha^{14}=2^{7} \mathrm{e}^{\frac{\mathrm{i} 14 \pi}{3}}=128\left[\mathrm{e}^{\frac{\mathrm{i} 2 \pi}{3}}\right] \Rightarrow \beta^{14}=128\left[\mathrm{e}^{-\frac{\mathrm{i} 2 \pi}{3}}\right] \\
& \alpha^{14}+\beta^{14}=128(2) \cos \left(\frac{2 \pi}{3}\right)=-128
\end{aligned}
\]
.
