Prove that $\bar{F} = r^n \bar{r}$ is conservative and find the scalar potential $\phi$ such that $F = grad\,\phi$.

Question

Prove that $\bar{F} = r^n \bar{r}$ is conservative and find the scalar potential $\phi$ such that $F = grad\,\phi$.

I can solve the 1st part by showing $curl\,F=0.$. I am facing two problems:

1. Nothing is said here about $n$. Is it an integer or any real number? If $n$ is a negative integer, then $\bar{F}$ becomes undefined at $r=0$ and hence $curl\,\bar{F}$. Please modify the question in such a general way that all cases are incorporated.
2. How to find the potential $\phi$. Is this $\frac{r^{n+2}}{n+2}$? If it is correct, then for $n=-2$, the potential is undefined.

What should be the modified question that incorporates all cases?