Are you asking for an analogy to a2 - b2 = (a -b)(a + b) or a3-b3 (n is integer)? I don't have an answer in this case. But if your problem stems from machine calculations, producing null for a not equal to b and n close to zero, then the working remedy might be: an - bn = an (1 - (b/a)n).
Thanks Dr Marek for your response, I, however, want to have an expression for an-bn which could some how simplify a term in an equation containing multiple parameters (non-linear eqn). I have tried solving the the equations by various software, but the outcomes are disobeying the experimental values in every instance. This, eventually led me to simplify the equations and which, finally, brings this problem.
This is my trick: A^n= \sum_{X=1}^{A} (X^n-(X-1)^n) if A,n Integers. Where I call M_n=(X^n-(X-1)^n) the complicate integer modulus. For A =Rationals use x=X/K, then you can arrange the Sum as a Step Sum, with Rational Step (or index) x=1/K, 2/K etc... . K must divide A rational so A=P/Q, then K=Q or one of its multiple. Here the example for A= P/2 n=2, K=Q=2: A^2=(P/2)^2= \sum:{1/2}^{P/2}2x/2-1/2^2. For a generic n,k there is a M_{n,K} develope. for K going to infinite you've the integral. here why FLT works for n=2.
The series in powers of z of exp(zlog(a)) converges slow for small a. In fact one need to take more than m=e/zlog(a)/ of terms of the series to quarantee the remainder tend to zero (/x/ means the absolute value of x).
if a and b larger than zero and n is a fraction we can make the change: a^n = c, b^n = d and we have a^n - b^n = c - d = (c^(1/2) - d(1/2)) (c^(1/2) + d^(1/2)) = (a^(n/2) - b(n/2)) (a^(n/2) + b^(n/2)) by the same way if n is a Root of rank m