Easy fact: If an ideal is a prime power, then its radical is prime.
I'd like to give a counterexample to the converse. A good candidate is $I=(y^2,xy) \subseteq K[x,y]$, since this post shows that it has prime radical: Is it true that an ideal is primary iff its radical is prime?
How do I show that $I$ is not a prime power? I am not sure where to begin; I've done the following: suppose $I=p^n$ for some prime ideal $p$. I try localising at $p$: $P_p^n=(y^2,xy)_p=(y)_p^2+(x)_p(y)_p$. What next?
Aside: is there an easier counterexample?