Why does natural units technique only works in equations in Physics?

Question

Example where it will not work is $(\frac{A}{B})^m = n$. Set $A=B=1$ and then solve for $m$. And example where it will work is:

$(E/c)^2 = p^2+ (mc)^2$. You can drop $c$ and put it back later by dimensional analysis. What is so special with equations in Physics why it always work?

Answer 1

It works not only for natural units, for all the sciences if you reconvert the magnitude to his original units, multiplying for the units in the original system that you did 1, using every unit converted to 1 to get the correct dimensions. There is only a way to do it