What is 0 to the power of 0?

1 Answer
Jun 24, 2015

Answer:

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#