I've heard before that some index laws don't apply in some situations, like when the base is negative.

For example, I entered $y = (x^{1.1})^{1.2} - x^{1.1\times1.2}$ into Wolfram Alpha and the graph it displayed was not $y=0$, contrary to my expectations.

From what I understand, $(x^a)^b = x^{ab}$ only when $x > 0$ and $a, b \in \mathbb{N}$. Is that correct?

Are there similar restrictions on other index laws? Specifically,

$x^a \times x^b = x^{a+b}$

$x^a \div x^b = x^{a-b}$ ($x \neq 0 $)

$x^\frac{a}{b} =\sqrt[b]{x^a}$ ($b \neq 0 $)