It's possible, but not trivial. The idea is to construct a sequence 1, x, x^2, ..., x^n and encode it in a single number. Encoding of sequences and operations on them can be done using Gödel's beta-function. See Definition 3 the following on this page (one of the first Google results for "exponentiation Peano Arithmetic").

In general, every computable function is representable by a formula in Peano Arithmetic.