I'm looking for a proof to show when $p^q$ for $p,q \in \mathbb{Q}$ is again in $\mathbb{Q}$, without factoring. I'm not sure, if it's possible, given these two numbers to say if the result is again rational and if so, calculate the result efficiently as well.
Thanks!
On
Expanding slightly on Arthur's comment. Let $p,q\in \mathbb{Q}$ and let $p=\frac{r}{s}$ and $q=\frac{m}{n}$ where $\gcd(r,s)=1$ and $\gcd(m,n)=1$. Then $p^q\in \mathbb{Q}$ iff $\exists l,k\in\mathbb{Z}$ such that $r=l^n$ and $s=k^n$.
I think this can be proved without resorting to using a prime number decomposition of $r,s$ just by using $\gcd$ but I will have to think about it for a little bit. It's certainly easier to prove using decomposition.
it is not correct for eg: $$ 2\in \mathbb{Q},\frac{1}{2}\in \mathbb{Q}\;\;\;but :\sqrt{2}=2^{\frac{1}{2}}\notin \mathbb{Q}$$