Rational subrieg of [0,∞)

Question

What is the subrieg of $[0,\infty)$ generated by $\mathbb Q_{\geq0}$, with ordinary exponentiation and $0^0=1$? Which equations hold between its expressions? Is the canonical map from the rieg freely generated by the rig $\mathbb Q_{\geq0}$ injective?

First distinction

The nonnegative rationals themselves are not closed under exponentiation: $2^{1/2}$ is already irrational. The question concerns the closure under all rieg operations and the relations in that closure.

Next step

State the universal property of the rieg generated by the rig $\mathbb Q_{\geq0}$, then write its canonical evaluation map into $[0,\infty)$. To prove a relation is additional, exhibit two terms with the same real value and prove that their equality does not follow from the rieg axioms and the specified rational rig relations. Merely finding two equal real expressions does not establish this.

Original problem 7.0.3.