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It is known that $i^{25}=i$, but ${i^{25}=(i^{4})^{\frac{25}{4}}}=1$. So, We have contradiction.

Let me rephrase it: what kind of limitation should be on multiplication of degrees?

What do you expect me to clarify? There are 3 well-known formulas 1. ${a^n} {a^m}=a^{n+m}$ 2. ${a^n}: {a^m}=a^{n-m}$ 3. ${(a^{m})^{n}=a^{mn}}$, But without limitation (for every $n$ and every $m$) they work only if $a$ is real and greater than zero. These formulas are also work in some other cases. My question: what are limitation on these formulas in case if $a=i$?

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closed as unclear what you're asking by user223391, Ofir Schnabel, Servaes, N. F. Taussig, Harish Chandra Rajpoot Oct 27 '15 at 10:47

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Here is a question you might want to consider: what exactly is the definition of $z^{25/4}$? $\endgroup$ – Paul Sinclair Oct 27 '15 at 3:38
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There is no contradiction, only a misconception. When working with complex numbers, $1^{\frac{25}{4}}$ has multiple values, not just $1$. (Or, to put it more generally, real positive numbers to fractional exponents can return multiple complex values).

A correct manipulation would be $i^{25} = ({i^4})^6 \cdot i = 1^6 \cdot i = i$.

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  • $\begingroup$ @Dr.MV Thanks Mark, cheers. $\endgroup$ – Deepak Oct 27 '15 at 3:42
  • $\begingroup$ You're welcome. My pleasure. - Mark $\endgroup$ – Mark Viola Oct 27 '15 at 3:43
  • $\begingroup$ I know that for many years. It is exactly why I ask. Let me rephrase it: what kind of limitation should be on multiplication of degrees? $\endgroup$ – Victor Victorov Oct 27 '15 at 3:48
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When dealing with complex functions, you often run into multi-valued functions, because when you circle around a singularity, it will pick up some constant value. There are three ways in which this is often handled:

  1. You can restrict the function to domains, called branches, that do not completely circle a singularity.
  2. You can live with the concept of a "multi-valued" function, treating the function value as being a set of complex numbers instead of a single complex number, or
  3. You can build a layered extension of the complex numbers, called a Riemann surface, to act as the domain, so that the function picks up each of its different values on a different layer.

The answer your question depends on which approach you take:

  1. Multiplication of exponents only works in general if the combined exponent lies in the same branch.
  2. Multiplication of exponents always works, and your $i = 1$ example is not a problem because the actual value is a set that contains both.
  3. Multiplication of exponents always works, but your example is wrong because you are mixing layers.
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