Ok this is quite dumb and absurd but it goes like this:
$i^3 = (i^4)^{3/4}= 1^{3/4} = 1$
I am not very sure what the mistake is, maybe if this was true we contradict $(i^4)^{3/4} \neq (i^{3/4})^3$
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6Fractional powers of complex numbers do not immediately make sense; see for example here. (Or just remember that $1^{1/2}=\pm1$.) – user1729 Jul 17 '23 at 13:22
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3Does this answer your question? What are the Laws of Rational Exponents? also https://math.stackexchange.com/questions/438/why-sqrt-1-cdot-1-neq-sqrt-12?noredirect=1&lq=1, https://math.stackexchange.com/questions/3210/simple-complex-number-problem-1-1?noredirect=1&lq=1, .... – Matthew Towers Jul 17 '23 at 13:24
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Cf. this question – J. W. Tanner Jul 17 '23 at 13:25
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Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – user1176409 Jul 17 '23 at 13:26
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5You can find many other questions here illustrating how $(a^b)^c = a^{bc}$ fails for complex numbers. – GEdgar Jul 17 '23 at 13:30