# Powers with complex/negative bases

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

• I think the restriction a, b > 0 ensures that a^x and b^x are defined. If a was allowed to be negative, then a^(1/2) wouldn't be defined, for example. – Aegis Apr 18 '16 at 17:34
• So would the relationship still hold true if complex numbers were allowed? – W. G. Apr 18 '16 at 17:38

## 1 Answer

(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).

• I appreciate Aegis, he says: I think the restriction a, b > 0 ensures that a^x and b^x are defined. If a was allowed to be negative, then a^(1/2) wouldn't be defined, for example., in respond to original question. Aegis, Please, do consider my further extended question. – Shivanshu Gupta Apr 18 '16 at 17:46