Physical quantities commonly have an integer dimensions. When I say integer, I mean they expressed in base dimensions (\(L, M, T, I, \Theta, N, J\)), raised to the integer powers: volume is \(L^3\), acceleration is \(LT^{-2}\) and so on. Of course, choice of basic dimensions is somewhat subjective, so one can choose volume instead of length, so length will have noninteger dimension of \(V^{\frac13}\). But the common observation is that the standard ones are sort of "atoms" which are never divided.

However, non-integer physical dimensions exist. First one is actually a whole class---spectral density of voltage V/√Hz, current A/√Hz and so on. All them have \(T^{\frac12}\) in their dimension. Another one is Hildebrand solubility parameter with dimension of \(M^{\frac12}L^{-\frac12}T^{-1}\) and measured in √Pa.

While it was hard to find these above, I wonder... Do there exist any more? Can fractions other than \(\frac12\) or even irrational physical dimensions exist?