  We study partial actions of exact discrete groups on C*-algebras. We show
that the partial crossed product of a commutative C*-algebra by an exact
discrete group is nuclear whenever the full and reduced partial crossed
products coincide. This generalises a result by Matsumura in the context of
global actions. In general, we prove that a partial action of an exact discrete
group on a C*-algebra $A$ has Exel's approximation property if and only if the
full and reduced partial crossed products associated to the diagonal partial
action on $A\otimes_{\max} A^\mathrm{op}$ coincide. We apply our results to
show that the reduced semigroup C*-algebra $\mathrm{C}^*_{\lambda}(P)$ of a
submonoid of an exact discrete group is nuclear if the left regular
representation on $\ell^2(P)$ is an isomorphism between the full and reduced
C*-algebras. We also show that nuclearity is equivalent to the weak containment
property in the case of C*-algebras associated to separated graphs.
