Just curious. The Planck length is lpl = (hG/c^{3})^{1/2} = 10^{-33}cm And it seems intuitive that it's c cube because space has three dimensions for the action. But the Planck time is tpl = (hG/c^{5})^{1/2} = 10^{-43}s So is there some obvious physical reason why c is to the power of five here?
The Planck time is just the Planck length divided by c. This adds two powers of c because they are inside the square root.
I don't see how a power of 3/2 looks natural. The planck volume has c^{9/2}. Those odd factors just show how "unnatural" the SI-units (where c, h, G, k are not nice numbers) are in terms of fundamental physics.
Newton said Gmm'/r=energy. In natural units, that means G is a (length)^2. The hbar and c are just put in to get G in cm^2. This is always unique.
what does "space has 3 dimensions for the action"? I mean that it's kind of weird, we don't know whether at Planck scale you need more than 3 spatial dimensions, so it's not so intuitive... On the other hand, everything seems normal under what is called dimensional analysis... So you have some constants ([itex]G, c, \hbar [/itex]) and you want to build characteristic quantities out of them .... So for everything, you just write: [itex] [X]= [c]^{a} [\hbar]^{b} [G]^{d} [/itex] and you solve for [itex]a,b,d[/itex]