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The iteration of exponents laws I learned gives: $$i^{4/3}=i^{4\cdot1/3}=(i^{4})^{1/3}=1^{1/3}=1$$ I know this is wrong but now the iteration of exponents laws I have used over the years somehow doesn't apply or is flawed. This is quite embarrassing as I have been studying mathematics for quite some time now. Please help.

MathR
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  • Taking non-integer powers of complex numbers requires some care in definition and handling - generally there are multiple possible values, and in many contexts it is not possible to make a consistent choice for the whole of your domain of interest. – Mark Bennet Feb 20 '21 at 12:27
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    In $\mathbb{C}$, the familiar law of indices that you learn for positive reals are mostly lost. $a^{bc}$ in general is not $(a^b)^c$ unless $b,c$ are nonzero integers. – user10354138 Feb 20 '21 at 12:27
  • $1^{1/n}=1$ is not true in general, in $\Bbb C$ you have $n$ different roots of unity. Also, working with square roots always requires some extra attentions (even in $\Bbb R$) – Ottavio Feb 20 '21 at 12:31
  • @Mathemagician314 this is funny – MathR Feb 20 '21 at 12:36
  • @user10354138 thank you, this is what I was looking for. So one has to use polar coordinates to calculate those kinds of expressions? – MathR Feb 20 '21 at 12:38
  • Fractional powers and roots of complex numbers is something I really, really discourage people from using, at least until they are very comfortable with how complex numbers work. This is basically the reason. Using them when teaching complex numbers for the first time really does the students a grave disservice. – Arthur Feb 20 '21 at 12:38
  • @MathR what do you mean? – Mathemagician314 Feb 20 '21 at 18:24
  • @Mathemagician314 What you uave written is wrong, very wrong. – MathR Feb 25 '21 at 20:48

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