  Let $F$ be a non-Archimedean locally compact field. Let $\sigma$ and $\tau$
be finite-dimensional semisimple representations of the Weil-Deligne group of
$F$. We give strong upper and lower bounds for the Artin and Swan exponents of
$\sigma\otimes\tau$ in terms of those of $\sigma$ and $\tau$. We give a
different lower bound in terms of $\sigma\otimes\check\sigma$ and
$\tau\otimes\check\tau$. Using the Langlands correspondence, we obtain the
bounds for Rankin-Selberg exponents.
