Are complex numbers subject to different rules of math? 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?
 A: Here is another version of your argument, with some answers. 
Simply, such a rule does not apply to complex numbers in general. 
Regarding your test question, writing $i\sqrt{-1}$ makes little sense. When we write $\sqrt2$, we are using the convention "the positive number such that its square is $2$". When you want square roots of $-1$, there is no obvious way from this point of view to distinguish between $i$ and $-i$. What we usually do to avoid this problem is consider $i$ as a number with $i^2=-1$. 
A: The definition of $a^x$ in complex analysis is $\exp(x \log(a))$, where $\log$ is the multi-valued complex (natural) logarithm.  Thus if $\text{Log}(a)$ is one value of this logarithm, the others are $\text{Log}(a) + 2 \pi i n$ for arbitrary integers $n$.  
Now the identity $(ab)^x = a^x b^x$ is not necessarily true.  Instead
$$ (ab)^x = a^x b^x \exp(2 \pi i n x)$$
A: 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
