  For a cardinal $\mathfrak{a}$, let $\mathrm{fin}(\mathfrak{a})$ be the
cardinality of the set of all finite subsets of a set which is of cardinality
$\mathfrak{a}$. It is proved without the aid of the axiom of choice that for
all infinite cardinals $\mathfrak{a}$ and all natural numbers $n$, \[
2^{\mathrm{fin}(\mathfrak{a})^n}=2^{[\mathrm{fin}(\mathfrak{a})]^n}. \] On the
other hand, it is proved that the following statement is consistent with
$\mathsf{ZF}$: there exists an infinite cardinal $\mathfrak{a}$ such that \[
2^{\mathrm{fin}(\mathfrak{a})}<2^{\mathrm{fin}(\mathfrak{a})^2}<2^{\mathrm{fin}(\mathfrak{a})^3}<\dots<2^{\mathrm{fin}(\mathrm{fin}(\mathfrak{a}))}.
\]
