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? 

This is a fun question and really scratches the mind. First of all we need to understand the term "undefined". Most people would think of it as an inconsistency in math, which is understandable because that is what we usually associate it with and which is usually shown as an error on a calculator. But it's not an inconsistency. Undefined simply means the answer isn't defined or described by a single number, but can rather be any possible number in our range of infinite numbers, and we therefore say that it is not defined or undefined. Sounds weird, I know, but keep reading. 
We know this by looking at the equation:
$$x \times 0 = 0$$ 
No mater what number $x$ is, the answer will always be equal to $0$. Therefore $x$ can be any possible number, because the answer will always be $0$. Cool right? Take the zero over and we get $x = 0 \div 0$ which justifies our statement of $x$ being undefined. Substitute that into your original equation and we get  $x \times 0 = 0$. Therefore $0 \div 0 \times0$ will be equal to zero.
However move the original equation to $0\times0\div0$, follow it right to left and we see the answer is undefined (being any possible number). This leads us to believe that the problem in the two equations lay in placement. Pretty strange for a single term equation, but that's how the gravy boat floats. 
But for your original equation of $0 \div 0 \times 0$ the answer is $0$. I'm not saying I'm correct, in fact my statement may be complete hullabaloo, but the moral of the story is this: don't be afraid of undefined. It's fun and is pretty awesome part of our abstract idea of math :)
