Could this be further generalized to n integers raised to nth power added together can have a number raised to nth power integer, hence why there is no relation

Could this be further generalized to n integers raised to nth power added together can have a number raised to nth power integer, hence why there is no relationship such that x^3 + y^3 = z^3, but there are an infinite number of x^3 + y^3 + z^3 = a^3 (where z in this case can work as +-1).