I wonder if there is a name for for the set of numbers that are rational powers of positive rationals.
This is an interesting restricted subset of the algebraic numbers.
It is clear that this set includes all rationals, and is a multiplicative group, as
$(\frac{a}{b})^\frac{m}{n} (\frac{c}{d})^\frac{r}{r}=(\frac{a^{ms}c^{rn}}{c^{ms}d^{rn}})^{\frac{1}{ns}}$, whit: $a,b,c,d,m,n,r,s$ integers.
But is not closed under addition as for example, as while: $ 1 $ and $\sqrt{2}=2^{\frac{1}{2}}$ belongs to the set, it is impossible to find integers: $a,b,m,n$ such: $1+\sqrt{2}=(\frac{a}{b})^\frac{m}{n}$
Loosely speaking, it would include all fractions constructed with arbitrary integer indexed roots of integers, and its negatives: (taking into account also the negative result of the even indexed roots of positive numbers)