exponent addition rule The exponent rule says: $ n^a *n^b=n^{a+b}$ right?
But, exponents like $b^{\frac 12}$ have two answers,
or even more in the case of $b^{\frac 1{10}}$. So doesn't this
create a contradiction, for example according to the rule:
$$9^{\frac 12}*9^{\frac 12}=9^1=9$$
But, $9^{\frac 12}=3$ or $-3$
Implying:
$3*-3=-9$, which is not $9^1$
I hope this doesn't have some obvious explanation 
I'm missing, but how can the exponent addition rule be true then
 A: When we restrict ourselves to the reals, we commonly use the convention that an $n$th root of a number $x$ refers to the positive $n$th root (whenever that is in doubt—e.g., when $n$ is even).  Thus, for instance, $\sqrt{x}$ conventionally refers to the positive square root, and if we wish to give it a different meaning, we usually have to specify it explicitly.
As long as we adhere to that convention, the rule you describe holds.  We say that $9^{1/2}$, the square root of $9$, is $3$ and not $-3$, and thus $9^{1/2}\times 9^{1/2} = 3 \times 3 = 9$.
When we expand from the reals to the complex numbers, however, that convention goes away.  A number such as $\frac{-1+i\sqrt{3}}{2}$ has two square roots—$\frac{1+i\sqrt{3}}{2}$ and $\frac{-1-i\sqrt{3}}{2}$—and there is no basis for identifying one of them as the square root.  One might occasionally prefer the root with the lowest argument (that is, the angle the value makes with the positive real axis in the complex plane), but that is not universally enough observed to assume without explicit mention.
