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i want to ask about main relation on powers of numbers ,

let $a$ be a negative real number and let , x,y be positive real numbers ,

is the relation ,

$ (a^{x})^{y} = a^{xy} $ true ??

and why ??

what if x,y are both negative ?

any ideas ?

what are the conditions which make this relation true ? and how did we conclude these conditions ?

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How well do you understand multi-valued functions? –  Calvin Lin Mar 16 '13 at 1:00
Well, I can certainly say it isn't always the case, for if it were, then $$(-1)^{1/2}=(-1)^{2/4}=[(-1)^2]^{1/4}=1,$$which clearly isn't the case. My guess is that in cases like this, the numbers need to be relatively prime. –  anon271828 Mar 16 '13 at 1:00
@CalvinLin , i have no ex-experince with multi-valued functions . –  Maths Lover Mar 16 '13 at 1:05
If $\,0>a\in\Bbb R\;,\;\;x\in\Bbb R\,$ , then it may well be that $\,a^x\,$ is not even defined in the reals, let alone fulfills this or that identity. –  DonAntonio Mar 16 '13 at 1:06
Try this Wiki link - Failure of power and logarithm identities –  Calvin Lin Mar 16 '13 at 1:15
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