Atomic radius by dimensional analysis

[areyouserious]

Hi,
So according to the Thomas-Fermi theory, the formula for atomic radius is
R=Z^(-1/3)*(hbar ^2)/m*e^2. (in other words Z^(-1/3) times the Bohr radius)

While the total energy has a Z^(7/3) dependence.

I need to get this from dimensional analysis. I can get the dependence on h, m, and e easily, but the Z^(-1/3) is a mystery to me. I understand that the Z, being unitless, must be tacked on to another unit, and working backwards, I can see that it works out if I attach a Z term to h and to e, and attaching a Z^(1/3) term to m. But how can I justify this? Why only 1/3 power for m, but first power for e and h? Please help, I am completely stuck.
Thanks

 

[Vanadium 50]

OK, Z means what? It's the nuclear charge, as well as the number of electrons. So where might you see Z's? The combination Ze makes sense in a way Zh does not, no? You can also see Z's appear if you are counting something - like electrons.

I hope this points you in the right direction.