A radial field is described by $F=ar^n \hat r$ .
1) Find the values for n for which $F$ is solenoidal in regions where $r$ is not equal to zero.
2) Find the values of $n$ ($r$ not equal to zero) for which the field is irrotational.
A solenoidal field fulfills the condition that $\nabla\cdot F = 0$. My doubt is... if $F$ is a radial field, I must consider that $F = (x^2 + y^2 + z^2)^{n/2}$? So should I make the partial derivatives of the components of $F$ with respect to $x$, $y$, and $z$, and set them equal to zero to obtain the value of $n$?
There is expression of divergence operator in spherical coordinate (See https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates#Del_formula) : $$\nabla\cdot F=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2F_r)+\frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}F_\theta\sin\theta+\frac{1}{r\sin\theta}\frac{\partial}{\partial\varphi}F_\varphi,$$ where $F_r$, $F_\theta$ and $F_\varphi$ is defined to be : $F=F_r\hat{e}_r+F_\theta\hat{e}_\theta+F_\varphi\hat{e_\varphi}$. In your case, $F_r=ar^{n+1}$, $F_\theta=0$, $F_\varphi=0$ hence whole expression simplifies. Therefore $$\nabla\cdot F=\frac{1}{r^2}\frac{\partial}{\partial r}ar^{n+3}=a(n+3)r^n.$$ If you impose $n=-3$, then you get $F$ is solenoidal everywhere. And for the second problem, since your field is radial, for all $n$ the field is irrotational.