Related: physics.stackexchange.com/q/75651, physics.stackexchange.com/q/4471/25301, and physics.stackexchange.com/q/87207/25301.

Possible duplicate (at least, I started to write the same answer): physics.stackexchange.com/q/112959/44126

@rob I wouldn't call it a duplicate since the other question is specifically about prefactors (although your same answer applies well here too!)

@dushyanth - You think the coefficients are often integers because those are the ones you have encountered at your level. But, depending on your area of study, it's very usual to encounter fractional exponents. For example, the correlation length of a lattice system near a phase transition takes the form $\xi = ( T - T_c)^\nu$ where $\nu$ is a fractional exponent. On a 3d lattice it has been measured (and calculated) to be around $\nu \simeq 0.627$. Phase transition, in particular, is an area with a large number of fractional exponents.

Only for unitless quantities you can have fractional exponents. Hint the answer lies with dimensional analysis.

Another note: there are a ton with square roots, which are just $x^{1/2}$ in disguise. By extension, any time you have a formula with an integer exponent greater than one, a simple rearrangement of that formula will give you another with a root ($x = \frac{1}{2} at^2 \implies t = \sqrt{2x / a}$) and vice versa ($v = \sqrt{F_T / \mu} \implies F_T = v^2 \mu$).

I think there is distinction between rational and irrational exponents. Rational exponents are ok with physical quantities, but irrational ones aren't.

Just a note: The area of a circle has pi in it not because of any choice on our end. Aliens from another galaxy would agree with us on the area of a circle and the value of pi. It's because the circumference of a circle is the radius times $2\pi$ and the area is the integral of the circumference through the radius. There are no choices that made it this way

In fluid mechanics, thermodynamics and heat transfer there are several examples of non-integer exponents and coefficients. Many emperical formulas are this way.

You get an integer exponent when the result is proportional to more than one thing, and they happen to be the same. The area of an ellipse with axes $a$ and $b$ is $A = \pi{}ab$, and a circle is an ellipse where $a = b$.

Counter-question: why should powers ever be fractions? We don't live in a 2.876 dimension universe, after all.

Imagine it was $F=m^{0.3412}a$. I think we would all agree to ditch m and add a new concept (say n) such that $n=m^{0.3412}$. Then we'd have $F=na$. So essentially the reason the powers are integers, at least for power of 1, is simply that we chose our units that way. For higher powers, I believe it's repeated application of linear equations that make those equations.

units, dimensions and integrations, usually. r^3 expresses someting in 3 dimensions, v^2 expresses the fact the effect of v is cumulative...

Because we live in a universe with an integer number of dimensions, and study objects with an integer subset of those dimensions. Also, what @Inquisitive said.

$F$ is $ma$ because this is what Newton said so. Force is defined that way just to make calculations easier.

Seriously speaking: If there were a formula with an irrational exponent it would be much less likely to be discovered than another formula with a rational exponent. Physicists, like most humans, prefer simplicity (even if it's often the complicated kind).

Fractals involve formulas with irrational exponents.

0.123 and 1.43 are both rational

Lots of linear or quadratic "laws" in physics are just truncated Taylor expansions.

@RBarryYoung - How do you know it's an integral number of dimensions?