First, try 99^3 = 970299; therefore, it is not the least possible value.
The above number is a multiple of 99, so we can factor this out to find the least possible value. So we try to factor 99:
99 = 9 * 11
99 = 3^2 * 11^ 1
and 3^3 * 11^3 = 35937
The least possible value is 4^1 + 11^2 = 125. Its sum is 125 which is a perfect cube.