  We show that the binomial and related multiplicative character sums $$
\sum_{\stackrel{x=1}{(x,p)=1}}^{p^m} \chi (x^l(Ax^k +B)^w),\hspace{3ex}
\sum_{x=1}^{p^m} \chi_1 (x)\chi_2(Ax^k +B), $$ have a simple evaluation for
large enough $m$ (for $m\geq 2$ if $p\nmid ABk$).
