The equation you gave is a complicated form of essentially a linear equation. It could be solved in the form you've given it if you gave the additional information that M, L, and T were all incomensurable and that you were looking for a, b, c over the integers. But that would be needlessly difficult.
Instead of the equation you wrote, consider these three equations derived from it by separating the "units":
M^a + M^b + M^c == M^0
(L^3)^a (L^2)^c == L
(T^-2)^a (T^-1)^c == T^0
So we could solve for {a,b,c} in terms of {M,L,T} but the values of {M,L,T} don't matter. You're really talking about this linear system of equations we can be dervied from those above:
{a + b + c == 0,
3 a + 2 c == 1,
-2 a - c == 0}
This linear system of equations is the correct way to think of this problem and how you'd want to enter the problem into any calculator