What is 0^0?

Author(s):

Michael Huber and V. Frederick Rickey

Pick up a high school mathematics textbook today and you will see that 0⁰ is treated as an indeterminate form. For example, the following is taken from a current New York Regents text [6]:

We recall the rule for dividing powers with like bases:

x^a/x^b = x^a-b (x not equal to 0) (1)

If we do not require a > b, then a may be equal to b. When a = b:

x^a/x^b = x^a/x^a = x^a-a = x⁰
(2)

but

x^a / x^a = 1 (3)

Therefore, in order for x⁰ to be meaningful, we must make the following definition:

x⁰ = 1 (x not equal to 0) (4)

Since the definition x⁰ = 1 is based upon division, and division by 0 is not possible, we have stated that x is not equal to 0. Actually, the expression 0⁰ (0 to the zero power) is one of several indeterminate expressions in mathematics. It is not possible to assign a value to an indeterminate expression.