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?

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