I've got what I think is a proof, am wondering if I've made a mistake.
Proof by contradiction: Suppose $\sqrt{n}=a/b$, with $a$ and $b$ being integers and coprime meaning $a/b$ is rational. Square it, so $n=a^2/b^2$ and take the $b^2$ over:
$b^2 n = a^2$
The way I see it, since $a^2$ is taking the prime factors of $a$ squared, it is impossible for there to be a lone factor in it. Meaning $a^2$ could be $2\cdot 2\cdot 3\cdot 3$, but not something like $2\cdot 5$ or $2\cdot 2\cdot 3$.
But if we look at the left side of the equation, any $n$ which has a lone factor, would have to be accompanied by another of the same value in $b^2$. Otherwise $a^2$ would have a lone factor in it which we already showed is not possible. But if $a$ and $b$ both share a factor, they can't be coprime.
This (I think) shows that the square root of any integer $n$ which has a lone prime factor is irrational.