infinity^0=infinity^(1-1)=infinity/infinity equals any positive number.

It is usually taken to be indeterminate, loosely for the reason you have given.

Infinity is generally ambiguous, ergo unless Infinity has a value, then we cannot calculate infinity further. The best explanation we probably have right now is that infinity is a "number" that does not have a value but we do know it doesn't have an end. It is uncountable, to say. But in mathematical theory, I theorise that Infinity cannot be affected by numbers, at all. But at the other side of the coin, Numbers can be affect by infinity. Much like: 1/Infinity = 0 or 0/infinity = 0, as I can name more. Infinity + 1 = Infinity. Thus, the answer will always be infinity unless you're using infinity to affect numbers.

But then there is the notion countable or uncountable infinite sets...

 

Infinity does not have a value as it is not a 'number'. Very very informally you can treat it as if it were 'the biggest number possible', but you have to take great care.

 

Also, you don't need to theorise about how infinity acts on numbers, this system is already studied and is called the extended number line. It is not a particularly nice system; it is not a field or a ring (the system does not quite behave like 'numbers'), nor does it have the standard semigroup structure.

Edited April 5, 201510 yr by ajb

We know that a^0=a/a as it is exponently one less than a^1 - the number needed to multiply number x to equal a^1 where x equals any number.

 

This is why 0^0=any number can equal x. because with x^0 the question is what when mutiplied by a equals x^1 with 0 this can equal any number.

Your question states

What is infinity to the power 0?

 

Before you can discuss this question you need to present which rules you are invoking concerning multiplication / exponentiation and show that that infinity is one of those mathematical objects that obeys these rules.

Edited April 14, 201510 yr by studiot