TITLE: Are complex numbers subject to different rules of math?
QUESTION [6 upvotes]: From what I know, the rule to distribute exponents is like:
$$(a b)^x = a^x b^x$$
Thus, if $a = \sqrt 2$ and $b = \sqrt 3$, $ab = \sqrt 6$.
However, the imaginary unit $i = \sqrt{-1}$ has a different behavior, because if I take $a = -2$, $b = -3$ and $x = 1/2$:
$${[(-2) (-3)]}^{1/2} = (-2)^{1/2} (-3)^{1/2} = \sqrt{-2} \sqrt{-3} = i\sqrt 2 i \sqrt 3 = - \sqrt 6$$
Although, before I learned complex numbers, I thought,
$${[(-2) (-3)]}^{1/2} = [6]^{1/2} = \sqrt 6$$
What am I making wrong here and which is the right answer?

REPLY [2 votes]: When you move into the complex arena, non-integer exponents can produce multiple answers.  
The errors and contradictions come in when you assume that the answer is the principal value.  When you constrain yourself to positive reals raised to a real value, the principal answer is the only answer and thus the correct answer.  This makes the rules safe to use.
In the case of the $i\sqrt{-1}$ question is ambiguous because you can't tell whether the square root means the principal value (which is usually the convention) or if it should have a $\pm$ in front of it.
If you are aware of the multiple possibilities (by including a factor of $e^{i2\pi}$ in your base(s)), or can somehow qualify the range of your answers, the rules can be used.
Ced