The problem is show that if $I$ and $J$ are ideals in a commutative ring s.t. $1 \in I+J$, then $1 \in I^m+J^n$ for all natural numbers $m, n$ (Marcus Number Fields Ch3 Problem 7).
I can write $1 = a + b$ for $a \in I$ and $b \in J$. Considering $m=n$ for now, from the binomial theorem we have $$ 1^n = 1 \quad\in\quad I^n + J^n + IJ\sum_{k=0}^{n-1}{I^kJ^{n-1-k}}.$$ I am not sure where to go from here. If $IJ\sum{I^kJ^{n-1-k}}$ were contained in either $I^n$ or $J^n$ we would be done but this does not always hold.
