What is zero to the power of zero?

To answer this, we need to think about what powers are.

What are powers?

A power is a number written small, slightly above and to the right of something, like this: 10³. In this case, it means 10 times 10 times 10 - or "ten multiplied by itself three times". The same way, x⁴ = x times x times x times x.

Multiplying powers

What happens if you have to multiply 10² by 10³? You could do it by saying "10 squared is 100 and 10 cubed is 1,000, and if you times them together you get 100,000," just by counting the noughts. What you're really doing, even if you don't realise it, is using the multiplying powers rule.

10² x 10³ = (10 x 10) x (10 x 10 x 10) = 10 x 10 x 10 x 10 x 10 = 10⁵

When you multiply powers together, you add the numbers above* and leave the bottom number alone. x¹² times x⁹⁷ = x¹⁰⁹. Easy, eh?

Dividing powers

It is a well-established fact that maths was invented by the same pedantically-minded two-year-old that invented the German language. That's what makes it easy. If multiplying powers together makes you do an addition sum, it makes Logical Sense that if you divide powers, you have to do a take away. That's exactly what happens. Try it if you don't believe me:

10⁵ ÷ 10³ = 10².

Powers of 0

How can you multiply something by itself no times? Well, you can't, really. But with powers, you can. You can turn it into a division calculation:

10⁰ = 10¹ ÷ 10¹ = 10 ÷ 10 = 1.

This is true... as long as the number underneath - 10, in this case - is not exactly zero.

Why not?

The reason why not is, 0 to the power of anything** is 0. Zero multiplied by anything, even itself, is zero. So we have a bit of a conflict: zero to the power of anything is 0 and anything to the power of 0 is 1.

Now, when maths sees a conflict, it has several options. One is to decide arbitrarily that one rule is more important than the other. Another is to figure out which is more important. A third is to throw up its hands and say "It just doesn't work," which is precisely what it does in this case.

0⁰ is undefined, just the same as 0÷0. You can't do it, it breaks the calculator. There is no answer that makes sense. It's an excellent question, and a really disappointing answer.

* As long as the number below is the same
** Anything positive, strictly