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This is a very basic question but it's been driving me crazy for a while... I know that $i^3 = i^2*i = (-1)i = -i$

But why does the following reasoning give me the wrong answer? $i^3 = (i^2)^{3/2} = (-1)^{3/2} = (-1)^{1/2} = i$

Thanks

sharding4
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Suss
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  • my guess is because you are using exponent laws meant to give positive answers or that you forget that all numbers have two sqrts and one of them for -1 is -i. –  Aug 05 '17 at 13:28
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    You expect $a^{bc}=(a^b)^c$, but even with real numbers this is not always true. For example, $$-1=(-1)^{2/2}\color{red}\ne((-1)^2)^{1/2}=1^{1/2}=1$$ – Simply Beautiful Art Aug 05 '17 at 13:28
  • @RoddyMacPhee A number cannot equal multiple things. – Simply Beautiful Art Aug 05 '17 at 13:29
  • it's not there are two sqrts for all values that's why sqrt unless taken to mean the positive sqrt is not a function. if you believe a negative times a negative is a positive, you have to accept as one of the consequences that -1^2=1^2 for example. –  Aug 05 '17 at 13:30
  • @RoddyMacPhee Of course, but you were suggesting we interpret $1^{1/2}=\pm1$, which would then imply many false things. You can only choose one value, usually the positive. – Simply Beautiful Art Aug 05 '17 at 13:33
  • Oh yes, and welcome to math.SE: since you are new, some basic information about writing math may be found here, here, here and here. I'd also recommend checking here and here – Simply Beautiful Art Aug 05 '17 at 13:41
  • Search tip: https://math.stackexchange.com/questions/tagged/complex-numbers?sort=frequent&pageSize=50 – Hans Lundmark Aug 05 '17 at 14:15
  • Very similar: https://math.stackexchange.com/questions/611529/why-is-i3-the-complex-number-i-equal-to-i-instead-of-i – Hans Lundmark Aug 05 '17 at 14:18

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