Question

What is 0 to the power of 0?

Answer 1

This is actually a matter of debate. Some mathematicians say `0^0 = 1` and others say that it is undefined.

Explanation:

See the discussion on Wikipedia:
Exponentiation : Zero to the power of zero

Personally I like `0^0=1` and it works most of the time.

Here's one argument in favour of `0^0 = 1` ...

For any number `a in RR` the expressions `a^1`, `a^2`, etc. are well defined:

`a^1 = a`
`a^2 = a xx a`
`a^3 = a xx a xx a`
etc.

For any positive integer, `n`, `a^n` is the product of `n` instances of `a`.

So what about `a^0`?

By analogy, that's an empty product - the product of `0` instances of `a`. If we define the empty product as `1` then all sorts of things work well. It makes sense as `1` is the multiplicative identity. If we were talking about the empty sum, then the value `0` would be natural.

If we're happy with that, what about `0^0`?

If it's the empty product of `0` instances of `0`, then it is `1` too.

Unfortunately, if we look at fractional exponents, we get some nasty behaviour.

Consider `(2^-n)^(-1/n)` for `n = 1, 2, 3,...`

As `n -> oo`, `2^-n -> 0` and `-1/n -> 0`

so you would hope `(2^-n)^(-1/n) -> 0^0` as `n->oo`

but `(2^-n)^(-1/n) = 2` for all `n in { 1, 2, 3,... }`

So exponentiation behaves badly in the neighbourhood of `0`