@cazzie to be didactic, squaring leads to areas and cubing to volumes only when the original quantity has a unit of length associated with it. Taking x^n where x is a dimensionless (“pure”) number will yield another dimensionless number. If the quantity does have a unit associated with that is not a length one needs to remember to apply the exponent to that unit as well, be it electric charge or whatever.
I recall being told once that the ancient Greeks insisted on geometric interpretations for what we now express as algebraic equations and that this had the effect of preventing them from developing algebra much further than they did.
@SavoirFaire and @Zyx, that is interesting, I’ve never heard of that term before either.