Explanation

Put t = aᵐ⁻ⁿ, so that aⁿ⁻ᵐ = 1/t. The expression becomes 1/(t − 1) + 1/(1/t − 1) = 1/(t − 1) + t/(1 − t) = 1/(t − 1) − t/(t − 1) = (1 − t)/(t − 1) = −1. The simplification is valid for every admissible a, m, n with t ≠ 0, 1.