What is 0 to the power of 0? Is it 1, 0, or undefined?

Question

Anything to the zero-th power is zero, but zero to any power is one, so what does this mean for zero to the zero-th power?

Thanks. I had a typo: 0^n=0, whereas n^0=1.

Answer 1

Actually you have it backwards.

A number to the zeroth power is 1
n^0 = 1

Zero to any power is 0.
0^n = 0

That aside...

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 x->0 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.

Answer 2

Hey there!

Puzzling's answer is going in a right direction.

Suppose you have the following true statements:

n^0=1
0^n=0

In the first statement, n^0=1, let n approach 0. As n gets closer and closer to 0, the value of n^0 is closer and closer to 1.

This is a limit, which can be expressed by the following:

lim n^0 =1
n->0
lim n^n=1
n->0+

However, the actual value of 0^0 is neither. It is indeterminate, meaning not yet determined.

The difference between indeterminate and undefined is by the following:

1/0 is undefined.
0/0 is indeterminate.

1/0 is infinity, but it has no defined value.

0/0 is not infinity. We are not sure of the answer. It can be 1, 2, 5/6, infinity etc.

0^0 is indeterminate, but expressed in a limit, it can have many values.

Like the following:

lim 0^n=0
n->0

It all depends on how we look at it.

Hope it helps!

Answer 3

You things mixed up,

0 to any power is 0 except 0 itself.
Any real number (including 0) to the zeroth power is 1

Answer 4

it would be undefined
but x^0=1 and 0^x=0

Answer 5

In algebra, it's undefined.

In calculus, it sometimes has a meaningful value. If the expression is really f(x)^g(x) where f and g both go to zero for some value of x, then there's a version of l'Hopital's rule lets you determine a value for it.