How to simplify a quotient with exponents (using power rules)

Question

I'm trying to simplify the following expression

(b (1 - x)^n (-1 + x)^-n (-1 + x - w x)^n)/(n (-1 + w) (1 - x + w x)).

All parameters and the variable x should be positive real numbers. As far as I can see, this can't be simplified further with Simplify or FullSimplify. I know that the simplified expression is

(b*(1 + (-1 + w)*x)^(-1 + n))/(n*(-1 + w))

but I'd like to know how to 'automate' this kind of simplification with Mathematica's functions, if possible.

Answer 1

expr = (b (1 - x)^n (-1 + x)^-n (-1 + x - w x)^n)/(n (-1 + w) (1 - x + w x));

Your stated assumptions including your comment

assume = {Thread[Variables@Level[expr, {-1}] > 0], Element[n, Integers]} // 
  Flatten

{b > 0, n > 0, w > 0, x > 0, n \[Element] Integers}

The simplified expression is

expr // Simplify[#, assume] &

(b (1 + (-1 + w) x)^(-1 + n))/(n (-1 + w))

Comparing with your presumed simplification

expr == b/(n (1 - w)) (1 - (1 - x + w x)^(n - 1)) // Simplify[#, assume] &

False

Consequently, the presumed simplification is not equal to the original expression under the given assumptions.