If $I$ is an ideal of a ring $R$, what is the meaning of $I^k$?
1) Is it the collection of $k-$ tuples of elements of $I$?
2) Or is it the collection of finite sums of $k$ products of elements of $I$? (That is $I^k=\{\sum_{r=1}^n i_{1r}i_{2r}...i_{kr}| i_{jr}\in I, n\in \mathbb{N}\})$
If $I$ is generated by the set $\{ a_i \}_{i \in I}$, then $I^2$ is the ideal generated by the set $\{ a_i a_j \, | \, i,j \in I\}$. For instance, $(a,b)^2 = (a^2,ab,b^2)$. In general, $I^k$ is the ideal generated by $$ \left\{ \left. \prod_{j=1}^k a_{i_j}\, \right| \, i_1,\cdots,i_k \in I \right\}. $$ Another example : $$ (a,b,c)^3 = (a^3, a^2b,ab^2,b^3,b^2c,bc^2,c^3,ac^2,a^2c,abc). $$ So you do get all finite sums of all products of $k$ elements in your original ideal. Note that the collection of all products of $k$ elements is not an ideal.
Hope that helps,