I'll bring in a less technical answer. Actually, even more dominant than integer exponents, is exponent 1, i.e., linear dependence between causes and effects. And often, exponent 2 comes from integrating this effect (i.e. kinetic energy), or multiplying two linear effects in which the same cause appears.

Why linearity? Well, this is as easy as 1 and 1 is 2, I'd say! This is simply the quite general principle that in the absence of a very specific interaction, doubling the intensity of a cause that has some effect will double the effect — e.g., twice as much inertia for twice as much mass.

I'll bring in a less technical answer. Actually, even more dominant than integer exponents, is exponent 1, i.e., linear dependence between causes and effects. And often, exponent 2 comes from integrating this effect (i.e. kinetic energy), or multiplying two linear effects in which the same cause appears.

Why linearity? Well, this is as easy as 1 and 1 is 2, I'd say! This is simply the quite general principle that in the absence of a very specific interaction, doubling the intensity of a cause that has some effect will double the effect — e.g., twice as much inertia for twice as much mass.

Edit: the excellent and detailed answer by Orion is expanding on this.