# Why can't you have zero to the power of zero?

##### 2 Answers

This is a really good question. In general, and in most situations, mathematicians define

But that is the short answer. This question has been debated since the time of Euler (i.e. hundreds of years.)

We know that any nonzero number raised to the

And that zero raised to a nonzero number equals

Sometime

Two source I used are:

http://mathforum.org/dr.math/faq/faq.0.to.0.power.html

Well, you kind of could have

The problem (if it is a problem) is that they don't agree on what the definition should be.

**Consideration 1:**

For any number **other than**

This is actually a definition of what the zero exponent mean. It's a definition chosen for good reasons. (And it doesn't "break" arithmetic.)

Here's one of the good reasons: defining

For example,

So what about

Cancellation (reducing the fraction) gives us **define**

So, maybe we should use the same rule to define

But . . .

**Consideration 2**

For any positive exponent, **not** a definition, but a fact we can prove.)

So if it's true for positive exponants, maybe we should extend it to the **define**

**Consideration 3**

We have looked at the expressions:

Now look at the expression

graph{y=x^x [-1.307, 3.018, -0.06, 2.103]}

One of the things you may notice about this, is that when

In some fields in mathematics, this is good reason to **define**

**Final notes**

Definition is important and powerful, but cannot be used carelessly. I mentioned "breaking arithmetic". Any attempt to **define** division so that division by

Last note: the definitions of