Show that the radical of the ideal is equal $ \langle X,Y\rangle $

Question

$\def\Rad{\operatorname{Rad}}$ Could you give me some hints how I can solve the followig exercise?

Show that the $\Rad(I)$ of the ideal $I=\langle X^5,Y^3\rangle $ of the ring $\mathbb{C}[X,Y]$ is $\Rad(I)=\langle X,Y\rangle $.

Answer 1

Clearly the radical of $\langle X^5, Y^3\rangle$ contains $X,Y$, and it does not contain $1$.

But now, $\langle X, Y\rangle$ is a maximal ideal contained in the radical of $I$, so it must be equal to it.