Radical of the ideal $I^2$ is a maximal ideal.

Question

Let $R=k[x,y]$, where $k$ is a field. Let us consider the ideal $I$ of the ring $R$ as $I=(x,y)$. Now consider $r(I^2)$, where $r(I^2)=\{x \in R: x^n \in I^2 \>\> for \>\>some\>\>n>0\}$. $\\$ Then show that $r(I^2)$ is a maximal ideal. $\\ \\$ To show that $r(I^2)$ is a maximal ideal, we need to show that $R/r(I^2)$ is a field. Please help me to show that is a Field.

Answer 1

Hint $$I \subseteq r(I^2)$$

Can you show that $I$ is maximal? What does this say about $r(I^2)$?