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OK this is probably a simple question, but I can't seem to find a good answer. I know that in dimensional analysis the units on both sides of the equation must be equivalent; I suppose the simplest example is: $$velocity \ (m/s) = \frac{distance}{time}=\frac{m}{s}$$ My question is what happens to units in an empirically derived equation where a dimension is raised to a non-integer power? For example, an equation for the evaporation of a pool is given as: $$G_{vap}=A\lambda u_{w} \rho_{v} \frac{P_v}{P_a}\left(\frac{2}{u_w^2 D_p}\right)^{n_c}$$ where $A$ is area (m^2), $\lambda$ is a dimensionless constant, $u_w$ is the wind speed (m/s), $\rho_v$ is the density in kg/m^3, $P$ is the vapour and atmospheric density (Pa), $D_p$ is the diameter (m), and $n_c$ is a dimensionless quantity in this case equal to 0.18.

I know that $G_{vap}$ has the units $kg/s$, which makes sense based on the units outside of the brackets, but what about the units inside the brackets? Clearly $D_p$ does not have the units $m^{0.18}$, for example, but why not? When is it not taken into account?

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When you raise things to a non-integral power you should imagine a constant inside to make it dimensionless. That $2$ in the numerator has units of $m^3/s^2$. If you change your units of measurement to feet and feet/hour the $2$ would change. Then the power has no units either and all is well.

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