I see lots of answers, and all quite nice and make sense but let’s approach this differently, for fun.
Most folks thinks in “addition land” fairly easily.
Let’s look at that first.When I write 3x =?
And take it to a number line, it means starting at the additive identity, zero, take steps of size +3, and take x steps.So it looks like this: (ex also shown: 3(5)=??)
Ok what about “multiplication world”?
When I write 9^x=?
And I take it to a number line, it’s new meaning is, starting at the MULTIPLICATIVE identity, one, take steps of size TIMES 9, and take x steps.
So it looks like this: (ex also show: 9^1.5)
Ok so then let us look at the problem being asked: 0^0
In the above context, in words, this value would be the value of starting at the multiplicative identity, one, and taking steps of size 0, and taking 0 steps, so clearly you never leave the 1. This would be analogous to 0(0) in add world, starting at 0, taking steps of size 0, 0 times. You never leave 0.
[learn more at : https://www.quora.com/How-come-0-0-equals-1-Who-decided-that ]