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I have the following expression in Mathematica:

$$\beta^{\frac{1}{(1 - \beta)}} - \beta^{\frac{\beta}{(1 - \beta)}}$$

I want to express it as

$$\beta^{\frac{\beta}{(1 - \beta)}}\left (\beta -1 \right)$$

I tried Factor and Simplify without success.

I would appreciate any advise.

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2 Answers 2

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expr = b^(1/(1 - b)) - b^(b/(1 - b));

Mathematica graphics

FullSimplify[expr,ComplexityFunction -> (Count[#, Power[_, _?Negative], Infinity] &)]

Mathematica graphics

reference: advice-for-mathematica-as-mathematicians-aid

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  • $\begingroup$ How come this only works with Negative power, even when ?Negative isn't used, when an expression with positive power clearly exists. $\endgroup$
    – Feyre
    Jan 6, 2017 at 12:50
  • $\begingroup$ Thanks. It works well, but I don't quite understand the logic behind this. I would have to read the reference that you left. $\endgroup$
    – John Shin
    Jan 6, 2017 at 13:24
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You can use

Simplify[Simplify[b^(1/(1-b))-b^(b/(1-b))/.b->(a-1)/a]/.a->1/(1-b)]
(*(b-1)b^(b/(1-b))*)

where $b=\beta$.

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    $\begingroup$ More simply Simplify[expr /. b -> a + 1] /. a -> b - 1 $\endgroup$ Jan 6, 2017 at 21:53
  • $\begingroup$ @SimonWoods yeah, good catch! I just took a to be the exponent of b, but your solution does the trick too, and it's much cleaner! $\endgroup$ Jan 6, 2017 at 21:58

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