\(\begin{array}{l}(x-5)^{x+1}-(x-5)^{x+13}=0 \Leftrightarrow(x-5)^{x+1}\left[1-(x-5)^{12}\right]=0 \\ \Leftrightarrow \text { hoặc }(x-5)^{x+1}=0 \text {, hoặc } 1-(x-5)^{12}=0 \\ (x-5)^{x+1}=0 \Leftrightarrow\left\{\begin{array}{l}x-5=0 \\ x+1 \neq 0\end{array} \Leftrightarrow x=5 \quad \text { (Thiếu } x+1 \neq 0, \text { trù } 0,25 d \text { ) }\right. \\ 1-(x-5)^{12}=0 \Leftrightarrow(x-5)^{12}=1 \Leftrightarrow\left[\begin{array}{l}x-5=1 \\ x-5=-1\end{array} \Leftrightarrow\left[\begin{array}{l}x=6 \\ x=4\end{array} \text {. Vậy: } \mathrm{x}=4, \mathrm{x}=5, \mathrm{x}=6\right.\right. \\\end{array}\)