$8.427 \sqrt[4]{10+8 \sin ^{2} x}-\sqrt[4]{8 \cos ^{2} x-1}=1$.
## Solution.

Let $\sqrt[4]{10+8 \sin ^{2} x}=u, \sqrt[4]{8 \cos ^{2} x-1}=v$.

$$
\begin{aligned}
& \left\{\begin{array} { l } 
{ u - v = 1 , } \\
{ u ^ { 4 } + v ^ { 4 } = 1 7 }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ u - v = 1 , } \\
{ ( u ^ { 2 } - v ^ { 2 } ) ^ { 2 } + 2 u ^ { 2 } v ^ { 2 } = 1 7 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
u-v=1, \\
u^{2}+v^{2}+2 u v+2 u^{2} v^{2}=17
\end{array} \Leftrightarrow\right.\right.\right. \\
& \Leftrightarrow\left\{\begin{array} { l } 
{ u - v = 1 , } \\
{ ( u - v ) ^ { 2 } + 4 u v + 2 u ^ { 2 } v ^ { 2 } = 1 7 }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ u ^ { 2 } v ^ { 2 } + 2 u v - 8 = 0 , } \\
{ u - v = 1 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
{\left[\begin{array}{l}
u v=-4, \\
u v=2, \\
u-v=1
\end{array}\right.}
\end{array} \Leftrightarrow\right.\right.\right. \\
& \Leftrightarrow\left\{\begin{array} { l } 
{ u v = 2 , } \\
{ u - v = 1 }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ u = 2 , } \\
{ v = 1 }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ 1 0 + 8 \sin ^ { 2 } x = 1 6 , } \\
{ 8 \cos ^ { 2 } x - 1 = 1 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
4 \sin ^{2} x=3, \\
4 \cos ^{2} x=1
\end{array} \Leftrightarrow\right.\right.\right.\right. \\
& \Leftrightarrow 4 \cos ^{2} x=1 \Leftrightarrow x= \pm \frac{\pi}{3}+\pi k
\end{aligned}
$$

Answer: $\quad x=\frac{\pi}{3}(3 k- \pm 1)$.