**Authors:** Richard Beigel

**Abstract:**
Let *k* and *t* be positive integers. It is well known that
*k*^{1/t} is either a positive integer or an irrational
number. This can be proved easily from the unique factorization
theorem.

An interesting question is how much number theory is necessary in
order to prove this theorem. Maier and Niven and Floyd have presented
proofs of this theorem that use no facts about prime numbers. The
former use the division algorithm and induction to simplify and
generalize Steinhaus's proof of the case *k* = *t* = 2; the
latter uses the Euclidean algorithm to simplify and generalize
Sagher's proof of the case *t* = 2. When *t* = 2, Maier and
Niven's technique is especially interesting because it does not
explicitly use any number theory, only very basic inequalities.

In this paper, we also prove the theorem for all *k* and
*t*. Although our proof is more complicated than Floyd's, it has
the advantage of not explicitly using any number theory, so it can be
presented to a very general audience. Our proof is simpler than Maier
and Niven's.