\(\left(x-3\right)^{x+5}-\left(x-3\right)^{x-15}=0\)
\(\Leftrightarrow\left(x-3\right)^{x-15}\left[\left(x-3\right)^{20}-1\right]=0\)
\(\Rightarrow\left[{}\begin{matrix}\left(x-3\right)^{x-15}=0\\\left(x-3\right)^{20}-1=0\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x-3=0\\x-3=1\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=3\\x=4\end{matrix}\right.\)\(\Rightarrow x=4\)