When is $(a^b)^c$ not equal to $a^{bc}$?

Asked Jul 05 '20 at 06:08

Active Jul 05 '20 at 06:08

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In this dsp.SE answer, the subtle point that $e^{-j\omega t} \ne ({e^{j\omega}})^{-t}$ is made. At school I learned that $(a^b)^c = a^{bc}$, which is also supported by answers like this one.

When does it not work? Is it when $b c$ is a complex number? Or is it when $b c$ is not an integer? Or both? The linked dsp.SE answer hints at these, but I am looking for the technically strict version of the rule with all the gotchas I should be aware of.

Also, is there a name for this rule?

asked Jul 05 '20 at 06:08

Milind R

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Whenever you write $a^b$ with $a$ not a positive real and $b$ not an integer, you are on shaky ground. – Angina Seng Jul 05 '20 at 06:11
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See this post and this post – Ben Grossmann Jul 05 '20 at 06:27
I had a similar question a while ago. – Bumblebee Jul 05 '20 at 07:05
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$(a^b)^c=a^{bc}\ $ holds whenever $\ a,b,c\ $ are real numbers and $a$ is positive. So, $\ bc\ $ need not be an integer. It does not hold in general in the complex numbers, which is the flaw in several fake proofs. – Peter Jul 05 '20 at 12:37
@Omnomnomnom thank you! – Milind R Jul 05 '20 at 13:18
@Peter Thanks, that's a pretty simple condition to keep in mind, but I get this feeling that it is slightly pessimistic... If you could write up an answer keeping the domain restriction of the numbers as minimal as possible, that would be great! – Milind R Jul 05 '20 at 13:25

When is $(a^b)^c$ not equal to $a^{bc}$?

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