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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 = nx^n-1
dx

is not valid for n = 1 at x = 0 unless 0⁰ = 1.
• The notation ∑ a_nxⁿ for polynomials and power series rely on defining 0⁰ = 1. Identities like:

1	=	∞ ∑ n=0	xⁿ when -1 < x < 1		e^x =	∞ ∑ n=0	xⁿ
1 - x							n!

and the binomial theorem

(1+x)ⁿ =

n
∑
k=0

(

n
k

)

x^k

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 a^b = e^b•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)

x^y

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 z^w is defined for nonzero z by choosing a branch of log z and setting z^w := e^{w log z}, but there is no branch of log z defined at z = 0, let alone in a neighborhood of 0.

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