What is zero to the power of zero?

Question

If you draw a graph of y = x^x and explore what happens as x approaches 0 we find that 0^0 =1. My calculator agrees, but is this true?

Answer 1

0^0 is undefined. For some purposes it may be useful to use the convention that 0^0=1 or 0^0=0, depending on what it is for. This question has plagued mathematicians for many years. I attended a seminar on this very question, and there was plenty of arguments cited for each case.

Answer 2

1

Answer 3

1

Answer 4

0^0 = 0^(1-1) = 0^1/0^1 = 0/0. So 0^0 cannot be done; it's indefinite. Also log 0^0 = 0 log 0 = 0 * (-infinity), which is indefinite.

However, you demonstrated with your calculator that lim (x -> 0) x^x = 1. You can prove this, perhaps by taking logarithms or by using the binomial formula. That doesn't mean that 0^0 = 1. If instead you take lim (x->0) x^(1/log(10)x), the result is 10, not 1, and substituting x = 0 directly gets you 0^(1/(-infinity)) which is 0^0. So if you use limits to define 0^0, you would also have to say 0^0 = 10, which contradicts 0^0 = 1. That is why 0^0 is left undefined.

Answer 5

"If you multiply xn by x, you obtain xn+1. So, the product of x0 and x is x [= x1]. If x is nonzero, x0 must therefore be equal to 1. Furthermore, we also have:

0^0 = 1

This seems to bother some people offhand (including a few textbook authors, who should know better), but x = 0 is not an exception to the above rule: It's indeed true that 0^0 = 1 . The most fundamental explanation is that an empty product [the product of no factor(s), which is what a zeroth power is] cannot possibly depend on the value of any factor since you are not using any such factor to form the "product". Thus, the value obtained for the zeroth power of any nonzero x must also be the correct value when x is zero."

Answer 6

0^0 is undefined. There is no defined value for this expression.
The rule that A^0=1 is only true if A=/=0. Remember that the exponent means the number of times that the base is multiplied by itself. 4^3 means three factors of 4, 4*4*4 which equals 64.
Because of the exponent rules, 4^0=1. So zero factors of a non-zero number is 1. But this doesn't make sense if the base is 0. So there is no definition for 0^0.

Answer 7

Yep, your calculator is telling you the truth. Any number, including zero (0) raised to the zero power is = 1.

Answer 8

yes 0^0 is 1

Answer 9

Hi. 1

Answer 10

You have to use differential calculus since you can't actually evaluate it at that point.

Let's say f(x) = x ^ x and g(x) = ln (x ^ x)

lim x --> 0 [x ^ x] = ?

lim x --> 0 [ f(x) ] = e ^ (lim x --> 0 [g(x) ] )

Now to find the limit of g(x) as x --> 0...

lim x --> 0 [ g(x) ] = lim x --> 0 [x (ln x) ]

= lim x -->0 [ ln (x) / (1 / x) ]

Now using L'Hopital's rule and taking the derivatives of the numerator and denominator of the expression

= lim x -->0 [ (1 / x) / (-1 / x^2) ]

= lim x -->0 [ (1 / x) / (-1 / x^2) ] = lim x --> 0 [ -x] = 0

And now plugging back in...

lim x --> 0 [ f(x) ] = e ^ (lim x --> 0 [g(x) ] ) = e ^ 0 = 1


So, the limit of x ^ x as x approaches 0 is 1.