My intuition is that this is true, because it makes sense for little sets (with cardinalities $|\mathbb{N}|,|2^\mathbb{N}|$). But i'm not sure at all that this is actually true.
If it's not, what's the real relation between $|B^A|$ and $|B|,|A|$.
(Obviously talking about infinite sets)
Surely not true for finite sets. Counterexample
$$A=\{1,2\} \text{ and } B = \{1,2,3\}.$$
And if Singular cardinals hypothesis is assumed, the result is again wrong as
$$\text{if } \vert \mathcal P(A) \vert \lt \vert B \vert \text{ and } \text{cf} \vert B \vert \le \vert A \vert \text{ then } \vert B^A \vert = \vert B \vert^+$$