Dividing by Zero Using Exponents I am sorry if I am missing some fundamental rule disproving my question, but I am very confused here. 

So $0^0 = 1$

and

$x/x = x^0 = 1$

so if $x = 0$

$0/0 = 1$

The problem here is people have told me 0 divided by 0 equals undefined, but by using exponents I have proved this wrong. I have to be doing something wrong here. So, what is wrong with what I just did?
Your reasoning is wrong beginning with the line "so if $x=0$", because division by $0$ is undefined. It doesn't matters whether you use a variable or a composite expression as the denominator, as long as it equals $0$, the argument is incorrect.

As is almost always the case with this type of questions, you seem to ignore the fact that mathematics is a deductive science. We begin from axioms and definitions, then demonstrate that some results logically follow from them. The properties of mathematical objects are not known due to observation of the physical world and are not laws of nature, but consequence of their definitions and the logical framework in which they are made (search for "construction of the real numbers".

It makes sense, however, to ask why for all $x \in R$, $x/0$ is undefined. The answer is in short, because it can't be defined meaningfully (in a way that doesn't contradicts the axioms of the real numbers).

Also, it is incorrect that "0 divided by 0 equals undefined". In ordinary treatments of mathematics, it doesn't equals anything. "is" has more than one meaning, not just the same as "equals". However, in formal treatments of mathematics like Metamath (link is to relevant theorem) the value of a function outside its domain (that is it, where it is not defined) equals an arbitrary object.
0^0 is not equal to 1. It is not defined. 
however:
(Tending to 0+)^(tending to 0) is 1. This is a standard limit. Thus what you are using is wrong.