Sorry for the lack of top-notch mathematical formatting, but I’ve done my best to make everything still look good.

00 is an interesting case when it comes to exponents. Whilst anything to the power of 0 is 1, raising 0 to any power gives 0. So what happens when you raise 0 to the power of 0? Is it 0 or 1?

Showing that anything raised to the power of 0 equals one is a fairly trivial matter. From the knowledge that xa ÷ xb = xa-b, it follows that xa ÷ xa = xa-a = x0. And since xa ÷ xa equals 1, x0 must equal 1.

But this doesn’t work if x is 0, because that results in 0a ÷ 0a = 0/0, which is either undefined or infinity (depending on your personal opinions).

As far as I can tell, 00 = 1. One of the more intuitive ways of seeing this is to look at the graph of y = xx.

Note how, as x approaches 0, y approaches 00 as well as 1.

If this doesn’t convince you, here’s a more formal proof.

The binomial theorem states that (a+b)n equals the sum of nCr × an-r× bn for r from 0 to n. You might know one particular form of this, that (a+b)2 = a2 + 2ab + b2, however it is technically a2b0 + 2a1b1 + a0b2. This leads to something interesting:

If we let a = 1, b = 0, and n = 2, then the left hand side becomes (1+0)2 = 12 = 1. But the right hand side becomes 12×00 + 2×11×01 + 10×02 and since we know that 01 = 02 = 0, it further becomes 12×00 = 1×00 = 00. And since the left hand side is 1, 00 must be 1.

This works for other values of n, even 0. The left hand side will always simplify to 1, and the right hand side will have every term except 1n×00 eliminated, as this always becomes just 00.

In the case of n = 0, the left hand is still 1 as (1+0)0 = 10 = 1, and  the right hand side becomes simply 10× 00 = 1 × 00 = 00.

From this it seem logical to conclude that 00 must be equal to 1.

There might be equations that only work if 00 has some other value, but I don’t believe I’ve encountered a situation like that. If you know of one, don’t hesitate to mention it in the comments.