Why Physical Power Laws Usually Have Rational Exponents

Many physical dependencies are described by power laws $$y=A\cdot x^a$$ , for some exponent a. This makes perfect sense: in many cases, there are no preferred measuring units for the corresponding quantities, so the form of the dependence should not change if we simply replace the original unit with a different one. It is known that such invariance implies a power law. Interestingly, not all exponents are possible in physical dependencies: in most cases, we have power laws with rational exponents. In this paper, we explain the ubiquity of rational exponents by taking into account that in many case, there is also no preferred starting point for the corresponding quantities, so the form of the dependence should also not change if we use a different starting point.