8.101. $\sqrt{2} \cos x+\cos 2 x+\cos 4 x=0$.
Solution.

Rewrite the equation as

$$
\begin{aligned}
& \sqrt{2} \cos x+2 \cos \frac{2 x+4 x}{2} \cos \frac{2 x-4 x}{2}=0 \Leftrightarrow \sqrt{2} \cos x+ \\
& +2 \cos 3 x \cos x=0 \Leftrightarrow \cos x(\sqrt{2}+2 \cos 3 x)=0
\end{aligned}
$$

From this,

1) $\cos x=0, x_{1}=\frac{\pi}{2}+\pi k=\frac{\pi}{2}(2 k+1), k \in Z$;
2) $\sqrt{2}+2 \cos 3 x=0, \cos 3 x=-\frac{\sqrt{2}}{2}, 3 x= \pm \frac{3}{4} \pi+2 \pi n$,

$x_{2}= \pm \frac{\pi}{4}+\frac{2}{3} \pi n=\frac{\pi}{12}(8 n \pm 3), k, n \in Z$.

Answer: $x_{1}=\frac{\pi}{2}(2 k+1) ; x_{2}=\frac{\pi}{12}(8 n \pm 3), k, n \in Z$.