system
Given the following post, respond as if you're commenting on Reddit:
human
Why do physical relationships tend to have small integer exponents?

Are these values just approximations, where a given phenomenon could be better modeled by a more complex relationship?  For example, could e=mc^2 actually be modeled more accurately e=m^1.02 *c^1.998, or some other such non-integer values?
gpt
One strong constraint is that the units have to work out. Consider E = m^1.02 c^1.998 . E has units of kg m^2 / s^2 , while m^1.02 c^1.998 has units of kg^1.02 m^1.998 / s^1.998. Since the units don't match up, it is meaningless to equate these quantities and this equation could never be a physical law. However, this doesn't completely rule out weird exponents: you could imagine equations like E = (1/2) m v^3.168 / c^1.618, which does have the right units.

Some small integer exponents are the way they are because we live in an integer number of spatial dimensions. For example, in d spatial dimensions, gravitational and electric forces obey 1/r^(d-1) laws. For d=3, this gives the familiar 1/r^2 laws.