$$\{\sqrt[n]{q}\;\mid\;n\in\mathbb N,\;n>0,\;q\in\mathbb Q,\;q>0\}$$ $$=\{2^{e_2}\cdot3^{e_3}\cdot5^{e_5}\cdot7^{e_7}\cdot11^{e_{11}}\cdots\;\mid\;e_i\in\mathbb Q,\;\text{finitely many }e_i\neq0\}$$
Multiplication in this group is just addition of the exponents for each prime. We can also define the GCD of any two elements by taking the minimum of the two exponents for each prime, and the LCM by taking the maximum of the exponents.
Is there a standard name or symbol for this group? I'm thinking of something like "the rational-radicals", $\sqrt[\mathbb N]{\mathbb Q}$ (though that suggests it includes all roots of unity).
For any positive integer $n$, and any Abelian group $G$, the $n$th power ($G\ni g\mapsto g^n\in G$) is a homomorphism, so the pre-image of any subgroup $S$ is also a subgroup, which could be denoted $\sqrt[n]S$. The union of all of these (varying $n$) is also a subgroup, which could be denoted $\sqrt[\mathbb N]S$. Here we can take $G$ to be the positive real numbers (or just the algebraic ones), and $S$ to be the positive rational numbers.