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This is merely a definition, and can't be proved via standard algebra. However, two examples of places where it is convenient to assume this:

1) The binomial formula: $(x+y)^n=sum_{k=0}^n {nchoose k}x^ky^{n-k}$. When you set $y=0$ (or $x=0$) you'll get a term of $0^0$ in the sum, which should be equal to 1 for the formula to work.

2) If $A,B$ are finite sets, then the set of all functions from $B$ to $A$, denoted $A^B$, is of cardinality $|A|^{|B|}$. When both $A$ and $B$ are the empty sets, there is still one function from $B$ to $A$, namely the empty function (a function is a collection of pairs satisfying some conditions; an empty collection is a legal function if the domain $B$ is empty).

$0^{0}$ is just one instance of an empty product, which means it is the multiplicative identity 1.

I'm surprised that no one has mentioned the IEEE standard for $0^0$. Many computer programs will give $0^0=1$ because of this. This isn't a mathematical answer per se, but it's worth pointing out because of the increasingly computational nature of modern mathematics, so that one doesn't run afoul of anything.

The use of positive integer exponents appears in arithmetic as a shorthand notation for repeated multiplication. The notation is then extended in algebra to the case of zero exponent. The justification for such an extension is algebraic. Furthermore, in abstract algebra, if $G$ is a multiplicative monoid with identity $e$, and $x$ is an element of $G$, then $x^0$ is defined to be $e$. Now, the set of real numbers with multiplication is precisely such a monoid with $e=1$. Therefore, in the most abstract algebraic setting, $0^0=1$.

Continuity of $x^y$ is irrelevant. While there are theorems that state that if $x_n to x$ and $y_n to y,$ then $(x_n + y_n) to x+y$ and $(x_n)(y_n) to xy$, there is no corresponding theorem that states that $(x_n)^{(y_n)} to x^y$. I don't know why people keep beating this straw man to conclude that $0^0$ can't or shouldn't be defined.

Maybe it's a good idea to put the problem into a broader perspective.

The minimum we need to define a power is a multiplicative semigroup, which basically means we have a set with an associative operation which we write as multiplication. If $S$ is the semigroup, the power function is defined as
$$Stimes mathbb Z^+ to S, (x,n) mapsto x^n =
begin{cases}
x & n=1\
x x^{n-1} & n>1
end{cases}$$
This power function has the fundamental properties
$$x^mx^n = x^{m+n},quad (x^m)^n = x^{mn}tag{*}$$
Note that up to now, we have used absolutely nothing about the elements of $S$ except for the fact that we can multiply them.

Now it is a natural question whether we can extend that definition from the positive integers to the non-negative integers, in other words, whether we can define $x^0$. Of course we would want to define it in a way that the power laws (*) still hold. This especially implies the following relations:
$$(x^0)^2 = x^0 tag{I}$$
(that is, $x^0$ is idempotent) and
$$x^0 x = x x^0 = x.tag{II}$$
Note that this does not imply that for any $xne y$, $x^0=y^0$.

Now for a given $S$ and a given $xin S$, there are three possible cases:

1. There does not exist an element $zin S$ that fulfils both (I) and (II). In that case, $x^0$ is undefined and undefinable without violating the power laws. For example, if you take $S$ as the set of positive even numbers, then $x^0$ is undefined for all $xin S$ (because there's no positive even number that fulfils either (I) or (II) for any x)


2. There exists exactly one element $zin S$ that fulfils those conditions. In that case, the only reasonable definition is $x^0 = z$. For example, for non-zero real numbers, you get $x^0=1$ this way.


3. 
   

   There exist more than one element $z_iin S$ fulfilling those equations. This is the case for $0^0$ because both $0$ and $1$ fulfil the equations. In that case, you have several choices:

   
   
   
   
   
   • You can select one of the possible values and define $x^0$ as that value. Of course you'd not randomly choose any value, but choose the one which is most useful. Which may be different in different contexts. Note that each of the choices gives a different valid power function.
   
   
   • You can leave $x^0$ undefined. In this case, $x^0$ is called unspecified because you could specify it (as in the previous bullet). In particular, the restriction of any of the power functions from the first bullet to $Ssetminus{x}$ will be the power function from this bullet.
   
   
   
   

Now one particularly interesting case is if you have a neutral element, that is, an element $ein S$ so that $ex=xe=x$ for all $xin S$. A semigroup with such an element is called a monoid.

It is easy to check that in this case, $e$ fulfils both (I) and (II) for any element $xin S$. Therefore you always get a valid power function by defining $x^0=e$. Indeed, by doing so, you get the general form
$$x^n = begin{cases}e & n=0\ x x^{n-1} & n>0end{cases}$$
which means that in a monoid, $x^0=e$ clearly is a distinguished, and therefore preferable choice of the power function, even for $x$ where other choices would be possible. Note that the real numbers form a monoid under multiplication, with $e=1$. Therefore this is a first hint that $0^0=1$ is a good definition.

The definition $0^0=1$ turns out to be useful also in other areas, for example when considering the linear structure (that is, the distributive law with addition), as it makes sure that e.g. the binomial formula
$$(a+b)^n = sum_{k=0}^n binom{n}{k} a^k b^{n-k}$$
also holds if any of $a$, $b$ or $a-b$ is $0$.

There's however one structure that doesn't favour $0^0=1$, and that is continuity: $x^y$ is discontinuous at $(0,0)$, and every value can be obtained as limit of a suitably chosen sequence of arguments approaching the origin. Therefore from the point of continuity, $0^0$ can be considered unspecified. However, since $x^y$ is discontinuous at the origin whether or not one defines it there, and independent of the value one chooses, that is not really an argument against choosing $0^0=1$, but more an argument against using that definition blindly.

I'm not aware of a context where the definition $0^0=0$ would be more useful, but I wouldn't want to exclude that it exists. Any other value of $0^0$ would violate condition (I) above, and therefore likely not be useful at all.

So in summary, the definition $0^0=1$ is the most useful in most situations, and not harmful in situations where one would otherwise leave $0^0$ unspecified, and if any context exists where another definition would be more useful, it's arather unusual context. Therefore the definition $0^0=1$ is the most reasonable one.

$0^0$ is undefined. It is an Indeterminate form.

You might want to look at this post.

Why is $1^{infty}$ considered to be an indeterminate form

As you said, $0^0$ has many possible interpretations and hence it is an indeterminate form.

For instance,

$displaystyle lim_{x rightarrow 0^{+}} x^{x} = 1$.

$displaystyle lim_{x rightarrow 0^{+}} 0^{x} = 0$.

$displaystyle lim_{x rightarrow 0^{-}} 0^{x} = $ not defined.

$displaystyle lim_{x rightarrow 0} x^{0} = 1$.

It's pretty straight forward to show that multiplying something by $x$ zero times leaves the number unchanged, regardless of the value of $x$, and thus $x^0$ is the identity element for all $x$, and thus equal to one.

For the same reason, the sum of any empty list is zero, and the product is one. This is when a product or sum of an empty list is applied to a number, it leaves it unchanged. Thus if the product $Pi()$ = 1, then we immediately see why $0! = 0^0 = 1$.

Without this property, one could prove that $2=3$, by the ruse that there are zero zeros in the product on the left (zero is after all, a legitimate count), and thus $2*0^0$, and since $0^0$ as indeterminate, could be 1.5, and thus $2=3$. I think not.

The approaches to $0^0$ by looking at $x^y$ from different directions, fails to realise that for even lines close to $x=0$, the line sharply sweeps up to 1 as it approaches $y=0$, and that the case for $x=0$, it may just be a case of not seeing it sweep up. On the other hand, looking from the other side, even in a diagonal line (ie $(ax)^x$), all do rapidly rise to 1, as x approaches 0. It's only when one approaches it from $0^x$ that you can't see it rising. So the evidence from the graph of $x^y$ is that $0^0$ is definitely 1, except when approached from $y=0$, when it appears to be zero.

"Everybody knows" that
$$
e^z = sum_{n=0}^infty frac{z^n}{n!},
$$
and when $z=0$ then the first term is $dfrac{0^0}{0!}$, so of course $0^0$ is $1$ since it's an empty product.

But it's also an indeterminate form because $displaystylelim_{xto z}f(x)^{g(x)}$ can be any positive number, or $0$ or $infty$, depending on which functions $f$ and $g$ are, if $f(x)$ and $g(x)$ both approach $0$ as $xto a$.

Knuth's answer is at least as good as any answer you're going to get here: http://arxiv.org/pdf/math/9205211v1.pdf See pp. 4-6, starting at the bottom of p. 4.

Another reason to define $0^0=1$ comes from probability, in particular Bernoulli trials.

These are independent repeated trials of an experiment, whose outcome is either positive or negative. So let $p$ be the probability of a success for each trial. Then the probability of exactly $k$ successes out of $n$ trials is $$p_k = {n choose k}p^k(1-p)^{n-k}.tag{B}$$ Now, suppose $p=1$. $(B)$ yields $p_n=0^0$. However, we already know that each trial will certainly occur, resulting in $p_k=0$ for all $k<n$ and $p_n=1$, whence $$0^0=1.$$

It depends on whether the 0 in the exponent is the real number 0, or the integer 0. These are two different objects and while the distinction is not often important, it is in this case.

Exponentiation by an integer has a universal specific meaning: Positive exponent is repeated multiplication, negative exponent is the inverse of repeated multiplication, zero exponent is the empty product, equal to the multiplicative identity denoted by 1. So if the exponent is the integer 0, then $0^0=1$ - the fact that this is an empty product with no terms trumps the fact that the terms which are not there are 0's (because they're not there).

Whereas exponentiation by a real or complex number is a messier concept, inspired by limits and continuity. So $0^0$ with a real 0 in the exponent is indeteriminate, because you get different results by taking the limit in different ways.

Note that all of the standard examples where it's "convenient" to have $0^0=1$ (e.g. power series) are all cases where the exponent is an integer.

Note that as long as the exponent is the integer 0, it doesn't matter if the base is the integer 0, the real 0, or pretty much any mathematical object on the planet for which multiplication is defined.

There are so many fantastic answers already, but no pictures which is sad. $0^0$ is undefined - is it the limited case of $x^0$, $0^x$ or $x^x$? The plots for $x^0$ (a line at 1) and $0^x$ (a line at zero) are boring, but $x^x$ is

_{Author/Site: Wolfram|Alpha}

_{Publisher: Wolfram Alpha LLC}

_{URL: https://www.wolframalpha.com/input/?i=plot+x%5Ex}

_{Retrieval date: 6/2/2015}

I don't always define $0^0$, but when I do, I define it as $1$.

Some indeterminates forms $0^{0}, displaystylefrac{0}{0}, 1^{infty}, infty − infty, displaystylefrac{infty}{infty}, 0 × infty, $ and $infty^{0}$

Futhermore,

$$lim_{ x rightarrow 0+ }x^{0}=1$$
and

$$lim_{ x rightarrow 0+ }0^{x}=0$$

See http://en.wikipedia.org/wiki/Indeterminate_form

Another approach and yet another result !

We define an exponential function on $mathbb{R}$ as a function $E:mathbb{R}rightarrow mathbb{R}$ such that
$$
E(x+y)=E(x)E(y) qquad forall x,y in mathbb{R}
$$
That's a very general and powerful definition, and I love it because captures the link between Lie algebras and Lie groups.

From this definition we have immediately that if $exists ain mathbb{R}$ such that $E(a)=0$ then $E(x)=0 ; forall x in mathbb{R}$. Nor continuity nor other topological properties are needed for this. In exponential fields theory such a function is called a trivial exponential function.

If $E$ is not this trivial function, it's easy to see that we must have $E(0)=1$ and, if $E(x)=1$ for some $x ne 0$ than $E(x)=1 forall x in mathbb{R}$, i.e. it is a constant (another trivial exp. function).

If $E$ is not trivial and $E(1)=a$ we have one particular funtion $E_a(n)$ and it is easy to see that $E_a(-1)=dfrac{1}{a}$ and $E_a(n)=a^n$ (and all other properties of integer or fractional exponents), so it is natural to write $E_a(x)=a^x$ .

Now we see that $0^x$ is the null trivial exponential function, and, since this function must be always null, we have $0^0=0$.

Take a look at WolframMathWorld's [1] discussion.

See if this gives you any clarification.

[1] Weisstein, Eric W. "Indeterminate." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Indeterminate.html

A paper for general public is published on Scribd : " Zero to the Zero-th Power" (pp.7-11) : http://www.scribd.com/JJacquelin/documents

Source: Understanding Exponents: Why does 0^0 = 1? (BetterExplained article)

A useful analogy to explain the exponent operator of the form $a cdot b^c$ is to make $a$ grow at the rate $b$ for time $c$.

Expanding on that analogy, $0^0$ can be interpreted as $1cdot0^0$ which is to say: grow $1$ at the rate of $0$ for time $0$. Since there is no growth (time is $0$), there is no change in the $1$ and the answer is $0^0=1$

Of course, this is just to grok and get an intuition or a feel for it. Science is provisional and so is math in certain areas. 0^0=1 is not always the most useful or relevant value at all times.

Using limits or calculus or binomial theorems doesn't really give you an intuition of why this is so, but I hope this post made you understand why it is so and make you feel it from your spleen.

A clear and intuitive answer can be provided by ZFC Set-Theory. As described in Enderton's 'Elements of Set Theory (available free for viewing here; see pdf-page 151): http://sistemas.fciencias.unam.mx/~lokylog/images/stories/Alexandria/Teoria%20de%20Conjuntos%20Basicos/Enderton%20H.B_Elements%20of%20Set%20Theory.pdf, the set of all functions from the empty set to the empty set consists merely of the empty function which is 1 function. Hence $0^0$ = 1.

Contents taken and reformatted from: The Math Forum

What is $0$ to the $0$ power?

This answer is adapted from an entry in the sci.math Frequently Asked Questions file, which is Copyright (c) 1994 Hans de Vreught (hdev@cp.tn.tudelft.nl).
According to some Calculus textbooks, $0^0$ is an "indeterminate form." What mathematicians mean by "indeterminate form" is that in some cases we think about it as having one value, and in other cases we think about it as having another.

When evaluating a limit of the form $0^0$, you need to know that limits of that form are "indeterminate forms," and that you need to use a special technique such as L'Hopital's rule to evaluate them. For instance, when evaluating the limit $sin(x)^x$ (which is $1$ as $x$ goes to $0$), we say it is equal to $x^x$ (since $sin(x)$ and $x$ go to $0$ at the same rate, i.e. limit as $xto0$ of $sin(x)/x$ is $1$). Then we can see from the graph of $x^x$ that its limit is $1$.

Other than the times when we want it to be indeterminate, $0^0 = 1$ seems to be the most useful choice for $0^0$. This convention allows us to extend definitions in different areas of mathematics that would otherwise require treating $0$ as a special case. Notice that $0^0$ is a discontinuity of the function $f(x,y) = x^y$, because no matter what number you assign to $0^0$, you can't make $x^y$ continuous at $(0,0)$, since the limit along the line $x=0$ is $0$, and the limit along the line $y=0$ is $1$.

This means that depending on the context where $0^0$ occurs, you might wish to substitute it with $1$, indeterminate or undefined/nonexistent.

Some people feel that giving a value to a function with an essential discontinuity at a point, such as $x^y$ at $(0,0)$, is an inelegant patch and should not be done. Others point out correctly that in mathematics, usefulness and consistency are very important, and that under these parameters $0^0 = 1$ is the natural choice.

The following is a list of reasons why $0^0$ should be $1$.

Rotando & Korn show that if $f$ and $g$ are real functions that vanish at the origin and are analytic at $0$ (infinitely differentiable is not sufficient), then $f(x)^{g(x)}$ approaches $1$ as $x$ approaches $0$ from the right.

From Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik):

Some textbooks leave the quantity 0^0 undefined, because the functions $0^x$ and $x^0$ have different limiting values when $x$ decreases to $0$. But this is a mistake. We must define $x^0=1$ for all $x$, if the binomial theorem is to be valid when $x=0$, $y=0$, and/or $x=-y$. The theorem is too important to be arbitrarily restricted! By contrast, the function $0^x$ is quite unimportant.

Published by Addison-Wesley, 2nd printing Dec, 1988.

As a rule of thumb, one can say that $0^0 = 1$, but $0.0^{0.0}$ is undefined, meaning that when approaching from a different direction there is no clearly predetermined value to assign to $0.0^{0.0}$ ; but Kahan has argued that $0.0^{0.0}$ should be $1$, because if $f(x)$, $g(x)$ $to0$ as $x$ approaches some limit, and $f(x)$ and $g(x)$ are analytic functions, then $f(x)^{g(x)}to1$.

The discussion of $0^0$ is very old. Euler argues for $0^0 = 1$ since $a^0 = 1$ for a not equal to $0$. The controversy raged throughout the nineteenth century, but was mainly conducted in the pages of the lesser journals: Grunert's Archiv and Schlomilch's Zeitshrift. Consensus has recently been built around setting the value of $0^0 = 1$.

References

Knuth. Two notes on notation. (AMM 99 no. 5 (May 1992), 403-422).

H. E. Vaughan. The expression '$0^0$'. Mathematics Teacher 63 (1970), pp.111-112.

Louis M. Rotando and Henry Korn. The Indeterminate Form $0^0$. Mathematics Magazine, Vol. 50, No. 1 (January 1977), pp. 41-42.

L. J. Paige. A note on indeterminate forms. American Mathematical Monthly, 61 (1954), 189-190; reprinted in the Mathematical Association of America's 1969 volume, Selected Papers on Calculus, pp. 210-211.

Baxley & Hayashi. A note on indeterminate forms. American Mathematical Monthly, 85 (1978), pp. 484-486.

Robert S. Fouch. On the definability of zero to the power zero. School Science and Mathematics 53, No. 9 (December 1953), pp. 693-696.

Let $$p=limlimits_{xto 0} x^x$$

Then
$$ln(p)=limlimits_{xto 0} xln(x)$$

$$ln(p)=limlimits_{xto 0} frac{ln(x)}{x^{-1}}$$

We can use L'Hopital on RHS as both numerator and denominator is tending to $infty$

$$ln(p)=limlimits_{xto 0} -frac{1/x}{x^{-2}}$$

This can be simplified by cancelling some x terms out.

$$ln(p)=limlimits_{xto 0} -x$$

Thereafter the limit can be applied

$$ln(p)=0$$

$$ p = e^0$$

Therefore $$p=1$$

The longstanding practice of leaving $0^0$ undefined is usually justified with arguments based on path dependent limits on the real numbers. As we see here, however, it is also possible to justify this practice based on purely discrete methods.

If the intuition of exponentiation on $N$ (where $0in N$) is to be repeated multiplication such that $x^2=xcdot x$, then we can formally justify the following definition:

1. $forall x,yin N: x^yin N$ (a binary function on $N$)




2. $forall xin N:(xne 0implies x^0=1)$




3. $forall x,yin N:x^{y+1}=x^ycdot x$

Here, $0^0$ is assumed to be a natural number, but no specific value is assigned to it.

From this definition, we can derive the usual Laws of Exopnents:

1. $forall x,y,zin N: (xne 0 implies x^y cdot x^z = x^{x+y})$




2. $forall x,y,zin N: (xne 0 implies (x^y)^z = x^{ycdot z})$




3. $forall x,y,zin N: (x,yne 0 implies (xcdot y)^z = x^zcdot y^z)$

For a detailed development based on formal proofs, see "Oh, the Ambiguity!" at my math blog.

Follow-up

A better approach, I have since found, is to construct a partial function on $Ntimes N$ with domain of definition $Ntimes N setminus (0,0)$ such that:

1. $forall x,y:[x,yin N land neg [x=0land y=0] implies x^yin N$




2. $forall x:[xin Nland xne 0implies x^0=1]$




3. $0^1=0$




4. $forall x,y:[x,yin N land neg [x=0land y=0] implies x^{y+1}=x^ycdot x]$

See my revised blog posting at above link (originally dated October 9, 2013).

There are several ways to prove that $0^0$ := 1. One way is to define the function of 2 variables f(x,y) = $x^y$. Then, evaluate the limit $lim_{(x,y)to(0,0)}x^y$. You can see - as I did it with Matlab - that $x^y$ goes to 1 when x $rightarrow$ 0 and y $rightarrow$ 0. Be aware that we cannot say $x^y$ equals 1 when x and y goes to 0, but $x^y$ approaches 1 as x and y goes to 0.

We must note that the term $0^0$ is not a normal number since there is no arithmetic calculation that leads to form the $0^0$. We have to options:

A) We set $0^0=0$ and demand that all arithmetic operations are subjected to number $0$ and not the term $0^0$, e.g. writing $0^0=0^{(1-1)}$ is forbidden. This definition is nice since adding zero doesn't change any equation.

$space$

B) We set $0^0=1$ and demand that all arithmetic operations are subjected to number $1$ and not the term $0^0$, e.g. writing $0^0=0^{(1-1)}$ is forbidden. This definition is nice since the limit $x^0 rightarrow 1$ as $x rightarrow 0$ is consistent with this definition.

Summa summarum: if we want to define the number $0^0$, then the most useful choice is $1$.