I'm working on a problem in Sharp's *Steps in Commutative Algebra*, to be precise exercise 4.28 which states the following:

Let $K$ be a field and $R = K[X,Y]$ be the polynomial ring in the indeterminates $X$ and $Y$ over $K$.
Let $I = (X^{3},XY)$.

(i) Show that the ideal $J\_{n}:= (X^{3},XY,Y^{n})$ is primary for each $n \in \mathbb{N}$.

(ii) Show that $I = (X)\cap (X^{3},Y)$ is a minimal primary decomposition of $I$.

(iii) Construct infinitely many different primary decompositions of $I$.

I don't have trouble sorting (ii) out as I found this quite easy. 
I do however struggle to show that $(X^{3},XY,Y^{n})$ is primary. I have tried doing it directly and also to look at the quotient ring $R/J\_{n}$ to show that every zero divisor is nilpotent. I also have not been able to explicitly find the radical of $J\_{n}$ in general. Any help will be appreciated, thanks in advance.