TITLE: Is there an algorithm that can tell whether the power of two rational numbers is rational?
QUESTION [6 upvotes]: It has been known since Pythagoras that 2^(1/2) is irrational. It is also obvious that 4^(1/2) is rational. There is also a fun proof that even the power of two irrational numbers can be rational.
Can you, in general, compute whether the power of two rational numbers is rational?
The reason I am asking, besides curiosity, is that the Fraction-type in Python always returns a float on exponentiation. If there is a quick way to tell if it could be accurately expressed as a fraction, the power function could conceivably only return floats when it has to.
EDIT:
By popular demand, I changed 0.5 to 1/2 to make it clearer that it is a fraction and not a float.

REPLY [16 votes]: One can do this much quicker than using prime factorization. Below I show how to reduce the problem to testing if an integer is a (specific) perfect power - i.e. an integer perfect power test.
Assume $\rm\:R,\: K/N\:$ and $\rm\:R^{K/N} = S\in\mathbb Q$, with $\rm\:K/N\:$ reduced, $\rm\:K,N\in\mathbb Z$. We show $\rm\:R^{1/N}\in \mathbb Q.\:$ By Bezout, $\rm\:gcd(N,K) = 1\:$ $\Rightarrow$ $\rm\:JN+IK = 1\:$ for some $\rm\:I,J\in \mathbb Z.\:$ Therefore
$$\rm 1/N \ =\ J + IK/N\ \Rightarrow\ R^{1/N}\ =\ R^{J+IK/N}\ =\ R^J(R^{K/N})^I = R^J S^I \in \mathbb Q$$
Conversely $\rm\:R^{1/N}\in \mathbb Q\ \Rightarrow\ R^{K/N} = (R^{1/N})^K\in \mathbb Q$.
So we've reduced the problem to determining if $\rm\:R^{1/N} = A/B  \in \mathbb Q$. If so then $\rm\: R = A^N/B^N\:$ and $\rm\:gcd(A,B)=1\:$ $\Rightarrow$ $\rm\:gcd(A^n,B^n) = 1,\:$ by unique factorization or Euclid's Lemma. By uniqueness of reduced fractions, this is true iff the lowest-terms numerator and denominator of $\rm\:R\:$ are both $\rm\:N'th\:$ powers of integers.
So we reduce to the problem of checking if an integer is a perfect power. This can be done very quickly, even in the general case, see D. J. Bernstein, Detecting powers in almost linear time. 1997.