# What is zero the the power of zero? (or 0^0)

The short answer is that some mathematicians think that 00 has a result of 1 or it is indeterminate. In mathematics, sometimes something only makes sense when you apply it to a problem in the real world. This is one of those problems. I personally would say that 00 is indeterminate because people still cannot agree on its result (I'm playing it safe). You could define it to equal 1 for convienience with some equations which is what some mathematicians do.
I will show you some of the reasons why there is a debate about this. The equations are simple, some are complex and cover different fields of mathematics so you should understand at least some of the reasons.

## Why some people think it equals 1

• Some people say that just because any other number to the power of 0 results in 1, then so should 0 but that does not mean it should, there is no proof in this statement.
• In differential calculus, the power rule:
 d xn = nxn-1 dx
is not valid for n = 1 at x = 0 unless 00 = 1.
• The notation ∑ anxn for polynomials and power series rely on defining 00 = 1. Identities like:
 1 = ∞ ∑  n=0 xn when -1 < x < 1 ex = ∞ ∑  n=0 xn 1 - x n!
and the binomial theorem
 (1+x)n = n ∑  k=0 ( n k ) xk
are not valid for x = 0 unless 00 = 1
• The combinatorial interpretation of 00 is the number of empty tuples of elements from the empty set. There is
exactly one empty tuple.
• Equivalently, the set-theoretic interpretation of 00 is the number of functions from the empty set to the empty set.
There is exactly one such function, the empty function.

## Why some people think it is indeterminate

• Since ab = eb•ln(a) to calculate 00 involves taking ln(0) which does not exist but this equation cannot be used to show that 02 = 0 either but it is true because 0×0 = 0.
• When 00 arises from a limit of the form:
 lim (x,y)→(0,0) xy
it must be handled as an indeterminate form
• Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form.
• In the complex domain, the function zw is defined for nonzero z by choosing a branch of log z and setting zw := ew log z, but there is no branch of log z defined at z = 0, let alone in a neighborhood of 0.