# 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

$(ab)^{x}&space;=a^{x}b^{x}$

• 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