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

Feb 26, 2015

This is a really good question. In general, and in most situations, mathematicians define ${0}^{0} = 1$.

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 $0$ power equals $1$
${n}^{0} = 1$

And that zero raised to a nonzero number equals $0$
${0}^{n} = 0$

Sometime ${0}^{0}$ is defined as indeterminate, that is in some cases it seems to be equal to $1$ and others $0.$

Two source I used are:
http://mathforum.org/dr.math/faq/faq.0.to.0.power.html

Mar 9, 2015

Well, you kind of could have ${0}^{0}$. In general, mathematicians leave ${0}^{0}$ undefined. There are 3 considerations that might lead someone to set a definition for ${0}^{0}$.
The problem (if it is a problem) is that they don't agree on what the definition should be.

Consideration 1:
For any number $p$ other than $0$, we have ${p}^{0} = 1$.

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 ${p}^{0}$ to be $1$ lets us keep (and extend) the rules for working with exponents,
For example, $\frac{{5}^{7}}{{5}^{3}} = {5}^{4}$ This works by cancellation and also by the rule $\frac{{p}^{n}}{{p}^{m}} = {p}^{n - m}$ for $n > m$.
So what about $\frac{{5}^{8}}{{5}^{8}}$?
Cancellation (reducing the fraction) gives us $1$. We get to keep our "subtract the exponents" rule if we define ${5}^{0}$ to be $1$.
So, maybe we should use the same rule to define ${0}^{0}$.
But . . .

Consideration 2
For any positive exponent, $p$, we have ${0}^{p} = 0$. (This is not a definition, but a fact we can prove.)
So if it's true for positive exponants, maybe we should extend it to the $0$ exponent and define ${0}^{0} = 0$.

Consideration 3
We have looked at the expressions: ${x}^{0}$ and ${0}^{x}$.
Now look at the expression ${x}^{x}$. Here's the graph of $y = {x}^{x}$:

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

One of the things you may notice about this, is that when $x$ is very close to $0$ (but still positive), ${x}^{x}$ is very close to $1$.

In some fields in mathematics, this is good reason to define ${0}^{0}$ to be $1$.

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 $0$ is allowed will break some important part of arithmetic. Any attempt.

Last note: the definitions of ${x}^{- n} = \frac{1}{{x}^{n}}$ and ${x}^{\frac{1}{n}} = \sqrt[n]{x}$ are also motivated in part, by a desire to keep our familiar rules for working with exponents.