What is $0\div0\cdot0$? We all know that multiplication is the inverse of division, and therefore
$x\div{x}\cdot{x}=x$
But what if $x=0$? $0\div0$ is undefined so $0\div0\cdot0$ should be too, but whatever happens when we divide that first $0$ by $0$ should be reversed when we multiply it by $0$ again, so what is the right answer? 0, undefined, or something else entirely? 

With the set of real numbers you have several operations. For example, you have addition ($+$) and division ($\div$). You can think of each operation as functions:
$$
\begin{align}
\mathbb{R}\times \mathbb{R} &\to \mathbb{R} \\
(a,b) &\mapsto a + b\\
\newline
\div: \mathbb{R}\times \mathbb{R}\color{red}{\setminus\{0\}} &\to \mathbb{R} \\
(a,b) &\mapsto a\div b
\end{align}
$$
The point with this is that you can add any two real numbers, but you can't divide any two numbers. The $\color{red}{\mathbb{R}}$ says exactly that. The denominator has to come from the real numbers $\mathbb{R}$, but can't be in the set consisting of $0$. The number is the denominator has to be non-zero. This is all a question about definition.
Because of this expressions like $0\div 0$ makes no sense. Neither does $7\div 0$. If $0\div 0$ doesn't make sense, then neither does any more complex expression if $0\div 0$ appears as part of that expression.
You indeed have a rules that says
$$
(x\div x) \cdot x = x,
$$
but this rule is only true for $x\neq 0$.
