Let’s have expressions that are finite-sized trees, with elements of $ \mathbb N$ as leaf nodes and operators and the operations {$ +,\times,-,/$ , ^} with their usual semantics as the internal nodes, with the special note that we allow arbitrary expressions as the right-hand side of the exponentiation operation. Are the equality between such nodes (or, equivalently, comparison to zero) decidable? Is the closure under these operations a subset of algebraic numbers or not?

This question is similar to this: Decidability of Equality of Radical Expressions but with the difference that here the exponentiation operator is symmetric in the type of the base and the exponent. That means that we could have exponents such as $ 3^\sqrt 2$ . It isn’t clear to me, whether allowing exponentiation with irrationals retains the algebraic closure.

This question is also similar to Computability of equality to zero for a simple language but the answers to that question focused on the transcendental properties of combinations of $ \pi$ and $ e$ , which I consider out of scope here.