Finding powers of ideals. Can any one assist me finding the product of ideals. 

Let $R=K[x_1,x_2,\dots,x_n]$, and $I=x_1R+x_2R+\dots+x_nR$. Then what is $I^m$? 

As $x_1^2+x_2^2$ belongs to $I^2$ it cannot be written as the product of two elements in $R$.

In general, if $I_1 = \sum_{i=1}^n x_i R$ and $I_2=\sum_{j=1}^m y_jR$, then $$I_1I_2 = \sum_{i=1}^n \sum_{j=1}^m x_iy_jR$$ is generated by the products of the generators of $I_1$ with the generators of $I_2$.
In your case it is $$I^n = \sum_{e_1+\dots + e_n=n} x_{e_1}x_{e_2}\cdots x_{e_n}R.$$ So if $R$ is the ring of polynomials in $x_1,\dots,x_n$, then $I^n$ is the ideal of polynomials whose monomials have total degree $\geq n$.
