Technically, 0^0 is an indeterminate form and has no specific solution. Accurate but unhelpful.
Practically, 0^0 highlights the issue that most of us don't have a good conceptual model for what exponents really do. How would you explain to a 10-year old why 3^0 = 1 beyond "it's necessary to make the algebra of powers work out".
to wrap my head around what exponents are really doing: some amount of growth (base) for some amount of time (power). This gives you a "multiplier effect". So, 3^0 means "3x growth for 0 seconds" which, being 0 seconds, changes nothing -- the multiplier is 1. "0x growth for 0 seconds" is also 1, since it was never applied. "0x growth for .00001 seconds" is 0, since a miniscule amount of obliteration still obliterates you.
Practically, 0^0 highlights the issue that most of us don't have a good conceptual model for what exponents really do. How would you explain to a 10-year old why 3^0 = 1 beyond "it's necessary to make the algebra of powers work out".
I use an "expand-o-tron" analogy
http://betterexplained.com/articles/understanding-exponents-...
to wrap my head around what exponents are really doing: some amount of growth (base) for some amount of time (power). This gives you a "multiplier effect". So, 3^0 means "3x growth for 0 seconds" which, being 0 seconds, changes nothing -- the multiplier is 1. "0x growth for 0 seconds" is also 1, since it was never applied. "0x growth for .00001 seconds" is 0, since a miniscule amount of obliteration still obliterates you.
This can even be extended to understand, intuitively, why i^i is a real number (http://betterexplained.com/articles/intuitive-understanding-...).