It is commonly taught that any number to the zero power is 1, and zero to any power is 0. But if that is the case, what is zero to the zero power?

Well, it is undefined (since x^{y} as a function of 2 variables is not continuous at the origin).

But if it could be defined, what “should” it be? 0 or 1?

**Presentation Suggestions:**

Take a poll to see what people think before you show them any of the reasons below.

**The Math Behind the Fact:**

We'll give several arguments to show that the answer “should” be 1.

- The alternating sum of binomial coefficients from the n-th row of Pascal's triangle is what you obtain by expanding (1-1)
^{n}using the binomial theorem, i.e., 0^{n}. But the alternating sum of the entries of every row except the top row is 0, since 0^{k}=0 for all k greater than 1. But the top row of Pascal's triangle contains a single 1, so its alternating sum is 1, which supports the notion that (1-1)^{0}=0^{0}if it were defined, should be 1. - The limit of x
^{x}as x tends to zero (from the right) is 1. In other words, if we want the x^{x}function to be right continuous at 0, we should define it to be 1. - The expression m
^{n}is the product of m with itself n times. Thus m^{0}, the “empty product”, should be 1 (no matter what m is). - Another way to view the expression m
^{n}is as the number of ways to map an n-element set to an m-element set. For instance, there are 9 ways to map a 2-element set to a 3-element set. There are NO ways to map a 2-element set to the empty set (hence 0^{2}=0). However, there is exactly one way to map the empty set to itself: use the identity map! Hence 0^{0}=1. - Here's an aesthetic reason. A power series is often compactly expressed as

SUM_{n=0 to INFINITY}a_{n}(x-c)^{n}.

We desire this expression to evaluate to a_{0}when x=c, but the n=0 term in the above expression is problematic at x=c. This can be fixed by separating the a_{0}term (not as nice) or by defining 0^{0}=1.

**How to Cite this Page:**

Su, Francis E., et al. “Zero to the Zero Power.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**Fun Fact suggested by**:

Francis Su