a: \(\left(x-1\right)^4+\left(y-3\right)^4=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x-1=0\\y-3=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=3\end{matrix}\right.\)
b: \(\left(x+y\right)^{2006}+2000\left|y-1\right|=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=0\\y-1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-1\\y=1\end{matrix}\right.\)
c: \(\left|x-y-5\right|+\left(y+3\right)^{2000}=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x-y=5\\y=-3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=y+5=-3+5=2\\y=-3\end{matrix}\right.\)