Could you elaborate on this? By "resolving the powers" do you mean magnitude of interacting forces? And by correct dimensions do you mean spatial dimensions?
By dimensions he means units. So taking into account the units of density/the gravitational constant (and any other pertinent quantities) and the units of the quantity you are calculating for you can derive an approximate formula just by looking at it and saying okay this unit needs ^2 and this one needs ^-1 and this one needs ^-3 for the end units to work out.
https://en.wikipedia.org/wiki/Dimensional_analysis