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Are the laws like the above one true? I used euler's formula.

r(eⁱˣ)=r(cos x + isin x)

and the equation says, this is true. Are there any exception?

Sanjay
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  • In general, no. See many posts here. For example https://math.stackexchange.com/q/1980763/442 – GEdgar Sep 25 '21 at 13:39

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Assuming $j$ and $s$ are integers, one indeed has ${(a^j)}^s={(a^s)}^j=a^{js}=a^{sj}$ for all $a\in\mathbb C$ (or $\mathbb C^*$ if you allow negative integer exponents, too).

If $j$ and $s$ are complex numbers though, things get trickier because the notation $a^j$ becomes ambiguous. When $a$ is a positive real number, the $j$-th power of $a$ is defined as $e^{j\ln(a)}$, however this is not possible in $\mathbb C$ because there does not exist a logarithm function that is continuous on $\mathbb C^*$.

KCJV
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  • Thanks. Does your argument also hold for rational numbers? Or just for the integers? – Sanjay Sep 25 '21 at 14:33
  • The same problem would arise for rationals. For instance, raising a number to the power $1/2$ would mean taking the square root, yet there also are not any continuous square root functions on $\mathbb C$ as a whole. – KCJV Sep 25 '21 at 15:49
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    Ohh... Got it... Thanks for explaining me :-) – Sanjay Sep 25 '21 at 16:04