I need help with this excercise

Let $k[X_1,\ldots,X_d]$ be the polynomial ring in $X_1,\ldots,X_d$ over a field $k$, and let $F_1,\ldots,F_m$ be forms of degree $n$. Assume that $(X_1,\ldots,X_d)=\sqrt{(F_1,\ldots,F_m)}$. Prove that $\overline{(F_1,\ldots,F_m)}=(X_1,\ldots,X_d)^n$.

Clearly, $(X_1,\ldots,X_d)^n\subseteq \overline{(F_1,\ldots,F_m)}$. But the converse...?

**Definition(Integral closure)**: Let $R$ be a ring and $I$ an ideal of $R$. An element $x$ is said to be integral over $I$ if $x$ satisﬁes a monic equation
$x^n + i_1x^{n−1} + ··· + i_n = 0$ such that $i_j ∈ I^j$ .The set of all elements that are integral over $I$ is called the integral closure of $I$,
and is denoted $\bar{I}$.