I've been searching about exponential rules on google and the first three results was these ones:
I didn't understand why they all say: $(a^b)^c=a^{bc}$. This is wrong, see for example: $((-2)^{2})^{1/2}=2\neq (-2)^1$.
Why does every site I read say this wrong rule? how can I reformulate this to make it true?
On
One way would be to simply require that $a>0$. Another way (working in the complex plane) would be to note that given $a\neq0$, both $(a^b)^c$ and $a^{bc}$ could have multiple values, but both must have some values (at least one) in common.
On
Remember that, in general, the exponential function is not injective. That is, you have to be careful about which solutions you're pulling out of the equation. Plus, if $a^x = a^y$ then you can't assume that $x = y$
Note that $((-2)^2)^{\frac12} = \pm 2$, one of those solutions remain when you apply the exponential rule. One way to think of it is that the exponential rule is always true, but you should be sure to check your solutions in the original equation.
On
\begin{align} (a^3)^4 & = \overbrace{(a^3)(a^3)(a^3)(a^3)}^{\large\text{4 factors}} = a^{\overbrace{3+3+3+3}^{\Large\text{4 terms}}} = a^{4\times3} \\[12pt] (a^3)^4 & = \underbrace{(aaa)(aaa)(aaa)(aaa)}_{\large\text{Adding four 3s yields } 4\times 3} = a^{4\times3} \end{align}
Reasonable textbooks will state that these things apply when $a\ge0$, and also apply when the exponents (in this case $3$ and $4$) are integers. Understanding what is displayed above should make it clear why the equalities hold when the exponents are nonnegative integers. Why they also hold when the exponents are negative integers is not hard to figure out.
Taking $n$th roots of negative numbers, however, is problematic; for example $25$ has two square roots: $\pm5.$
All these sites you found appear to be aimed at a beginning algebra audience. At this level, typically all bases are positive and all exponents are integers (at least to begin with). And with positive bases and integer exponents, it is indeed the case that $(a^m)^n = a^{mn}$. At this level, in the US at least, one should never expect to see something like $(-2)^{1/2}$ because that ventures into the realm of imaginary numbers, which is not typically covered until 2-3 semesters after beginning algebra.
All you really need is $a > 0$, then for any $m, n \in \Bbb R$, we have $(a^m)^n = a^{mn}$. If $a = 0$ then you have to worry about negative exponents because they'll lead to division by zero. If $a < 0$ then it is not true in general that $(a^m)^n = a^{mn}$, as you've already noted with your example.