Any dimensionless number multiplied by another dimensionless number is always another dimensionless number right?
Therefore any scale multiplied by a group of dimensionless numbers is also dimensionless.
A, B and C are arbitrary scales
x = (A/B)x' (x' is dimensionless variable),
y = Cy' (y' is dimensionless variable)
xy = (AC/B)x'y' (term is also dimensionless right?)