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Let's say I have a fraction like this:

(a b x^2 + a^3 x) / (a^2 b x)

and I would like to simplify it by multiplying both the numerator and the denominator with

1/x^4

while introducing the following:

a/x = A and b/x = B

So in the end I'd get:

(A B + A^3) / (A^2 B)

Is there a way to do this in Mathematica?

Thank you!

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To avoid potential naming conflicts, you should avoid using capital letters to start user-defined variables/functions (e.g., A and B). In the desired result a factor of A can be cancelled.

(a b x^2 + a^3 x)/(a^2 b x) /. {a -> A*x, b -> B*x} // Simplify // Together

(A^2 + B)/(A*B)

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