I've recently begun looking at complex surfaces, i.e. manifolds admitting a holomorphic structure, and this obviously includes computations involving the complex differential form $dz$. I'm comfortable with real differentials $dx$ and $dy$, and I understand that $dz = dx + idy$, and accordingly we get relations like $dx^2 + dy^2 = dzd\overline z$, $\mathrm{Im}(dz) = dy$, etc, but some computations involving $dz$ still seem to elude me.
In my reading, I've come across the following equation: $$\nu = \left| \mathrm{Im}\left(dz^{p/2}\right)\right| = \left| \mathrm{Im}\left(z^{(p-2)/2}\,dz\right)\right|$$ where $\nu$ is a measure on a surface $M$ defined on a collection of arcs. This appears to imply that $dz^{p/2} = z^{(p-2)/2} \, dz$. However, if we're applying the standard power rule of complex analysis, isn't this missing a constant factor of $\frac p 2$? That is, shouldn't we have $$ \left| \mathrm{Im}\left(dz^{p/2}\right)\right| = \left| \mathrm{Im}\left(\frac p 2 z^{(p-2)/2}\,dz\right)\right|? $$ Or are we ignoring a constant factor because we're looking at differential calculus and we can mod out by scalar multiples when considering length forms?