This is saying that "R" is a length so has units "L" (meters, feet, parsecs, etc.), v is a speed so has units "L/T" (T is seconds, minutes, hours, years, etc.), is a density so have units of "mass over volume" so "M/L^3" (M is grams, kg, slugs, etc. and, of course, volume is "length cubed"). is viscosity and so (as we all know!) has units of "M/LT". We don't know what function of each of these is involved but whatever function is involved, it can only change the "number" of each unit involved, and whether it is in the numerator or denominator so it must have some exponent (negative if in the denominator).

That is, whatever the function f is, we must have the result and, combining like bases,

Because drag has units of " , in order that those have the same units we must have a+ b- 3c- d= 1, -b-d= -2, and c+ d= -2.

Again, f can be any function. If it were functions like [itex]e^x[/itex] or sin(x) which have no units for the arguments, we must have constants involved that will cancel any units on x, and then constants multiplying the result of the functions having the correct units for the result. But if we expect simply multiplying or dividing (for some physical reason) then the factors must involve lengths, times, masses as shown.