Let $\displaystyle A$ be a commutative ring with unity.
If $\displaystyle M,N$ are distinct maximal ideals of $\displaystyle A$, then
(1)$\displaystyle M+N=A$.
(2)$\displaystyle M^a+N^b=A (a,b\ge1)$.
For (1), the sum of ideals is again an ideal (link). Thus, M+N is an ideal containing M. By hypthesis, M+N should properly contain an maximal ideal M. Thus M+N=A.
For (2), every proper ideal in A is contained in a maximal ideal in A and note that A contains the unity (link).
Assume $\displaystyle M^a+N^b , a,b\ge1$ is a proper ideal in A. Then, $\displaystyle M^a+N^b ,a,b\ge1$ should be contained in a maximal ideal. It follows that $\displaystyle M^a+N^b , a,b\ge1$ should be contained in either M or N (check their intersection). Contradiction !
Thus, $\displaystyle M^a+N^b , a,b\ge1$ is an ideal in A which is not a proper ideal in A. We conclude that $\displaystyle M^a+N^b=A ,a,b\ge1$.