If x can be a positive real number (for example a fraction with a numerator and denominator), then why does the following relationship hold true only if and only if a and b are strictly positive real numbers? In other words, why doesn't this relationship also hold true if a and b are complex or negative? See this link for power of product. https://proofwiki.org/wiki/Exponent_Combination_Laws
2026-04-11 18:04:19.1775930659
Powers with complex/negative bases
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(I AM COMMENTING IN THE ANSWER SECTION BECAUSE I DON'T HAVE 50 REPUTATION)
1) (2i)^2=-4 and ((2)^2).((i)^2)=4.(-1)=-4
2) (ii)^3=-1 as well as ((i)^3).(((i)^3)=-1
3) By partial deduction we can say (ab)^{x} =a^{x}b^{x} holds for for all a,b and c where a and b belongs to the set of complex number(Z) and x to R+.
4) I am extending his question: Prove or disprove the statement (by any method, most preferably the shortest one): we can say that (ab)^{x} =a^{x}b^{x} holds for for all a,b and c belonging to the set of complex number(Z).