Решите уравнение: (x-4)^4+(x-8)^4=32.

\[(x - {4)}^{4} + (x - 8)^{4} = 32\]

\[t = \frac{x - 4 + x - 8}{2} = \frac{2x - 12}{2} =\]

\[= x - 6\]

\[(t + 2)^{4} + (t - 2)^{4} = 32\]

\[2t^{2} + 48t^{2} = 0\]

\[2t^{2}\left( t^{2} + 24 \right) = 0\]

\[t^{2} = 0 \Longrightarrow t = 0.\]

\[x - 6 = 0\]

\[x = 6.\]

\[t^{2} + 24 = 0\]

\[t^{2} = - 24 \Longrightarrow \ \ \ нет\ решения.\]

\[Ответ:\ \ 6.\]