How to get rid of the denominator in the numerator and the denominator in mathematics

Question

I have the following expression

(-1 + 1/p)^B/(-1 + (-1 + 1/p)^(A + B))

How can I multiply the denominator and number point by p ^ (A + B), i.e. get rid of the denominators in both the numerator and the denominator? I tried varous Expand, Factor, Simplify, etc., but none of them worked.

Thank!

Answer 1

If I understand your question, you can teach Mma some algebra:

r = {(k__ + Power[a_, b_]) Power[c_, b_] -> (k Power[c, b] + Power[a c, b]),
      p_^(a_ + b_) q_^a_ -> p^b ( q p)^(a),
      (a_ + b_) c_ -> (a c + b c)
    }

and then define

s1 = ((-1 + 1/p)^B/(-1 + (-1 + 1/p)^(A + B)))

f[a_, c_] := (Numerator[a ] c //. r)/(Denominator[a ] c //. r)

So,

f[s1, p^(A + B)]  

there is

((1 - p)^B*p^A)/((1 - p)^(A + B) - p^(A + B))  

Answer 2

I have to say that I did not understand the original question. However, trying to understand the intriguing solution given by belisarius, I came up with the following:

expr = (-1 + 1/p)^B/(-1 + (-1 + 1/p)^(A + B));

Together@(PowerExpand@FunctionExpand@Numerator@expr/
 PowerExpand@FunctionExpand@Denominator@expr)

Conclusion (set by the velizari):

As an alternative:

PowerExpand@FunctionExpand@Numerator@expr/PowerExpand@
 FunctionExpand@Denominator@expr

gives

or

FunctionExpand@Numerator@expr/FunctionExpand@Denominator@expr

Thanks belisarius for another good lesson in Mma's power.

Answer 3

, p ^ (A + B),