Since we are talking about rational exponents, I assume it makes sense to restrict to positive real numbers. If I got it right the question is whether a set $S$ where no element is a product of some others to rational powers is necessarily countable.
One can show non-constructively that the answer is no, that is, that there exist uncountable sets with this property. This is because the condition on $S$ is equivalent to $\{\log x: x\in S\}$ being linearly independent over the rational numbers.
Since there is a basis for $\mathbb{R}$ as a vector space over $\mathbb{Q}$, and such a basis is necessarily uncountable, there is an uncountable set $S$ with the given property.