My input is:
$ \frac{1}{a^x}==(\frac{1}{a})^x $
I want to get output:
True
But getting output:
$ a^{-x}==\left(\frac{1}{a}\right)^x $
Question: How to get output: True
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2 Answers
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3
Your expression is only true if a is positive.
Simplify[a^-x == (1/a)^x, a > 0]
True
answered Nov 21, 2019 at 16:32
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3$\begingroup$ Is there a way to ask Mathematica in which cases it is true and in which not? So instead of me specifing for cases when
a > 0, Mathematica will give me all possible cases with correspondingTRUE/Falsevalues. $\endgroup$ Nov 21, 2019 at 16:40 -
1$\begingroup$ @vasili111 That is an excellent, yet different question. I strongly suggest asking it as a separate question! $\endgroup$– rubenvbNov 22, 2019 at 14:28
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$\begingroup$ @rubenvb Done: mathematica.stackexchange.com/questions/210131/… $\endgroup$ Nov 22, 2019 at 17:38
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1
You can use PowerExpand to find out the ratio in general:
PowerExpand[(1/a)^x a^x, Assumptions->True]
E^(2 I π x Floor[1/2 + Arg[a]/(2 π)])
For generic x this expression is only 1 when
Floor[1/2+Arg[a]/(2 π)] == 0
Using Reduce gives:
Reduce[Floor[1/2 + Arg[a]/(2 π)] == 0, a, Complexes]
(Im[a] != 0 && Re[a] < 0) || Re[a] >= 0
So the equality is only untrue for negative reals (i.e., along the branch cut for logs and powers).
Check with Simplify:
Simplify[a^-x == (1/a)^x, (Im[a] != 0 && Re[a] < 0) || Re[a] >= 0]
True
answered Nov 21, 2019 at 17:02