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

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.

• (1 - x)^n (-1 + x)^-n will only simplify to -1 if n is an integer. You might want to re-examine your assumptions. In any event, look up Assuming[]. – J. M.'s technical difficulties Jul 18 '15 at 16:56
• Yes, it's true that n is an integer -- sorry, should have added that. – jmbierna Jul 18 '15 at 17:02
• Are you sure about the simplified form? I get (b*(1 + (-1 + w)*x)^(-1 + n))/(n*(-1 + w)) – m_goldberg Jul 18 '15 at 18:10
• No, I've now realised that my 'simplified form' was indeed the wrong expression. I agree that your solution is correct (and see Answer below). Thanks for your help. [*** note I've edited the question above to correct it] – jmbierna Jul 18 '15 at 21:02

## 1 Answer

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.

• Yes, this is correct. I mistakenly pasted the wrong 'presumed simplification', and now I see that the result here is what I'm looking for. Thanks very much. [*** note I've edited the question above to correct it] – jmbierna Jul 18 '15 at 21:00