Why is $x^0 = 1$ except when $x = 0$?

Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.

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For non-zero bases and exponents, the relation $ x^a x^b = x^{a+b} $ holds. For this to make sense with an exponent of $ 0 $, $ x^0 $ needs to equal one. This gives you:

$ x^a \cdot 1 = x^a\cdot x^0 = x^{a+0} = x^a $

When the base is also zero, it’s not possible to define a value for $0^0$ because there is no value that is consistent with all the necessary constraints. For example, $0^x = 0$ and $x^0 = 1$ for all positive $x$, and $0^0$ can’t be consistent with both of these.

Another way to see that $0^0$ can’t have a reasonable definition is to look at the graph of $f(x,y) = x^y$ which is discontinuous around $(0,0)$. No chosen value for $0^0$ will avoid this discontinuity.

This is a question of definition, the question is “why does it make sense to define $x^0=1$ except when $x=0$?” or “How is this definition better than other definitions?”

The answer is that $x^a \cdot x^b = x^{a+b}$ is an excellent formula that makes a lot of sense (multiplying $a$ times and then multiplying $b$ times is the same as multiplying $a+b$ times) and which you can prove for $a$ and $b$ positive integers. So any sensible definition of $x^a$ for numbers $a$ which aren’t positive integers should still satisfy this identity. In particular, $x^0 \cdot x^b = x^{0+b} = x^b$; now if $x$ is not zero then you can cancel $x^b$ from both sides and get that $x^0 = 1$. But if $x=0$ then $x^b$ is zero and so this argument doesn’t tell you anything about what you should define $x^0$ to be.

A similar argument should convince you that when $x$ is not zero then $x^{-a}$ should be defined as $1/x^a$.

An argument using the related identity $(x^a)^b = x^{ab}$ should convince you that $x^{1/n}$ is taking the $n$th root.

If $a$ and $b$ are natural numbers, then $a^b$ is the number of ways you can make a sequence of length $b$ where each element in the sequence is chosen from a set of size $a$. You’re allowed replacements. For example $2^3$ is the number of 3 digit sequences where each digit is zero or $1$: $000, 001, 010, \ldots, 111.$

There is precisely one way to make a zero length sequence: one. So you’d expect $0^0=1$.

$$0^x = 0, \quad x^0=1$$

both are true when $x>0$.

What happens when $x=0$? It is undefined because there is no way to chose one definition over the other.

Some people define $0^0 = 1$ in their books, like Knuth, because $0^x$ is less ‘useful’ than $x^0$.

Exponents are only “basically” defined under the natural numbers above zero. By this I mean, defined as “iterated multiplication” the same way multiplication is defined as iterated addition.

The property $a^0 = 1$ only arises when we look at generalizing multiplication to the integers. We do this by:
\begin{align}
a^4 / a^3 &= (a\cdot a\cdot a\cdot a)/(a\cdot a\cdot a) = a^1\\
a^4 / a^3 &= a^{4-3} = a^1
\end{align}

And using this, we can say:

$$a^2 / a^3 = a^{-1} = 1/a$$

and also:
\begin{align}
a^2 / a^2 &= 1\\
a^2 / a^2 &= a^{2-2} = a^0 = 1
\end{align}

So we say $a^0 = 1$.

However, notice that these proofs don’t have any meaning when $a=0$, because the whole concept/idea involves fractions, and you cannot have zero be in the denominator.

When we say $2^0 = 1$, we really mean:

$$ 2^{1-1} = 2^1 / 2^1 = 2/2 = 1$$

But we cannot say the same for $0^0$:
$$0^{1-1} = 0^1/0^1=0/0=\text{UNDEFINED}$$

If we use the idea of set exponentiation to define exponentiation of cardinals, we have the following natural idea:

$$A^B:=\{f:B\to A\}$$

We define the exponential of cardinals as follows: $|A|^{|B|}:=|A^B|$. It’s easy to check that this agrees with our intuition for exponentiation of natural numbers when $B$ is nonempty.

There is only one set representing the cardinal $0$, namely the empty set. Then we may look at $0^0$ as follows:

$$0^0=|\emptyset|^{|\emptyset|}=|\emptyset^\emptyset|
=|\{f:\emptyset\to\emptyset\}|=1$$

The key point is to understand the meaning of multiplication. The $a \times n$ means that you add n a-items to zero rather than “together” because “together” leaves you with uncertainty when no items are taken. If we understand that 0 must be the answer when n=0, we understand what counting means: 0 is the “constant of integration” that lives in any empty set that we add your items into in order to count them. You start counting with 0. If you embed that into definition, a lot of confusion is resolved right away. So, instead of defining $$a \times n = \underbrace{a + a + \cdots + a}_n, $$ you define $$a \times n = \mathbf{0} \underbrace{+ a + a + \cdots + a}_n.$$ You note that 0 is also the additive identity. So, counting starts with additive identity.

Similarly, when you multiply n identical a-items together, what you actually do is you take the product of those items with 1, $a ^ n = \mathbf{1} \underbrace{\times a \times a \times \cdots \times a}_n.$

To repeat, the syntax that corresponds to the meaning of basic math operations must be: $$a \times n = \mathbf{0} \underbrace{+ a + a + \cdots + a}_n \;\;\;\text{and}\;\; a ^ n = \mathbf{1} \underbrace{\times a \times a \times \cdots \times a}_n$$ instead of $$a \times n = \underbrace{a + a + \cdots + a}_n\;\;\;\text{and}\;\;a ^ n = \underbrace{a \times a \times \cdots \times a}_n.$$

Now, if n = 0, everything but identities vanishes. You will have $$a\times n = 0 \;\;\;\text{and}\;\; a^0 = 1.$$ Note that “$\times nothing$” in “$1 (\times nothing)$” does not mean scale 1 with 0. It means that you take 1 alone and do not scale it with anything. When n=0, I do not “add” anything to my “constants of integration”. Even if the items that I ignore to “count” are of size a=0 or whatever, I still have
$$a\times n = 0 \;\;\;\text{and}\;\; a^0 = 1.$$ Making base zero does not change anything, if you know what is behind $a^n$ rather than blindly drill the rule $0^{anything} = 0.$ This is what my programmer’s mindset tells me. Might be I am too bold by viewing the product as the computation

function product(arguments)
  result = 1
  foreach arg in arguments
    result = result * arg

Yet, it was found intuitive. This is exactly what product (exponentiation) means. That is why I say that injecting the identity in front of the sums/products is a natural (i.e. true) thing to do and this must be reflected in the definition. I am happy that it meshes well with the combinatorial argument.

If you can explain how two definitions underpin each other or if you have any objection, please comment. I believe that all problems that arise in continuous analysis are because you do not have pure zeroes there. That is, formally, you may consider $\quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t^2}}\right)^t$ as $ 0^0$. But, I believe that actually you have infinitesimals there, which are not zeroes exactly. Consider the total contribution as a product of a items, whereby every contributes 1/a. If you start reducing the item contribution, $1/a \to 0$, total contribution is unchanged. However, if you really manage to ignore the contributions completely then you will get $0 \times \infty = 0$. You just do not care about the sizes of contributions that you do not count.

Similarly is with $1 \times a \times a \cdots$. If you cut all contributions, you are left with 1, regardless of magnitudes of a. If you receive result that is different from 1 then you have failed to zero all the contributions.

One of the definitions of the power $a^b$ is $e^{b \log a}$, where $$e^x=1+x+\frac{x^2}{2}+\dots $$

  • If $a$ is nonzero, and $b=0$, then $a^b=e^0=1$.
  • If $a=b=0$, the expression $b \log a=0 \log 0$ is an indeterminate form, thus $a^b$ is undefined. Nonetheless, in many applications we assign the indeterminate form $0 \log 0$ the limiting value $\lim_{x \to 0}x \log x=0$, so that $0^0$ is defined to be $1$ as well.

Because when you say ‘to the power of zero’ it just means that you are dividing that number by itself. Therefore zero to the power of zero is zero divided by zero which is undefined.