# What is 0^0?

On desmos, if f(x)=x^x, f(0)=1. The function's domain is also restricted to x>=0. I do not know why since -1^-1 is -1 and -2^-2 is -1/4.

In school, I was taught that x^0=1 and 0^x=0 so why is one overriding the other?

So why is x^x restricted to x>=0 and why does 0^0=1?

### 19 Answers

- 9 months ago
what is 1 to the power 2..1*1 ...so...1^2 =1

what is 2 to the power 3... 2*2*2 so 8

then what is 0 to the power 0?

it is 0

Physically speaking, it is nothing..i mean..it is saying like nothing has happened...i.e 0(nothing) raised to the power 0(nothing).

some call it indeterminate form also..

- RaymondLv 79 months ago
It is indeterminate (you cannot determine the value).

The rule for limits is that the limit must be the same, regardless of the "direction" you take to get there.

Bluntly, a limit must be unique.

As soon as you can show more than one limit value, then the value does not exist and the operation cannot be determined.

Let's look at n^0

and allow n to get closer and closer to zero (for example, n = 1/2, 1/4, 1/8...)

The value of n^0 will be 1 for all values of n, therefore the limit is "1"

If this is valid, then 0^0 = 1

Now consider 0^n

and allow n to get closer and closer to zero.

The value of 0^n will be 0 for all values of n, therefore the limit is "0"

If this is valid, then 0^0 = 0

Both operations are valid, yet they reach a different limit. Therefore the limit does not exist.

- Jim MoorLv 79 months ago
While the actual answer is "indefinite", I would think it would be agreed upon that the value sould be '1'

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- RicardoLv 49 months ago
1 = (y^x)/(y^x) = y^(x-x) = y^0

But I don't think y can be 0, cause 0^0 is undetermined.

If you calculate, for instance, .0001^.0001, you'll see the result is .99, but not 1.

- Φ² = Φ+1Lv 79 months ago
In school you were taught that 0^(-1) = 1/0 = 0 ?

We know that 1^1 = 1, so by binomial expansion

1 = 1^1

1 = (1+0)^1

1 = 1 * 1^1 * 0^0 + 1 * 1^0 * 0^1

1 = 1 * 1 * 0^0 + 1 * 1 * 0

1 = 0^0 + 0

then 0^0 = 1

This may not be strictly true, but it is at least a useful value. Like 0! = 1.

- ted sLv 79 months ago
0^0 = 0...but limit as x ---> 0+ of f(x) = x^x = 1...thus f(x) is not continuous at x = 0...and the proper definition of x^x is e^(x ln x )

- Jeffrey KLv 69 months ago
You must find the limit as x approaches 0 of f(x) = x^x. Use can use l'hopitals rule or just put in a small number for x and see what happens. If x=.001 then x^x is close to 1.

X can not be negative because of fractional negative values. If x = -1/2 then x^x = 1 / sqrt(-1/2) and you can not take the square root of a negative number.

- Anonymous9 months ago
Any number to the zeroth power is 1. At least, that's what I remember.

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