Your teacher didn't tell you (or told you, but you didn't recognize it as something valuable and forgot it) that exponentiation to zero is a new definition over exponentiation to positive integers. Exponentiation to positive integers is defined somehow, and that definition says nothing about exponentiation to zero. It is a new definition, not something that you deduce.
The same holds for 0^0 or 0/0 (with some amounts of confusion, lies, and hypocriticism).
Now, you probably now that this is because 0 and 1 are the identity elements of the additive and multiplicative groups of real numbers, respectively. Probably wouldn't've helped you as kid though
I think it's much more productive to teach children
2x0 = 2x1 + 2x(-1) = 2-2 = 0
and
2xx0 = 2xx1 x 2xx(-1) = 2/2 = 1
Inverting the concept and having those patterns stem from a fundamental identity can wait until its not seen as the mathematical equivalent of "Because I said so".
Another good way of thinking about it is that if 10^a 10^b = 10^(a+b), you can set a=3 and b=0 to get 10^3 10^0 = 10^3. If you divide both sides by 10^3, 10^0 = 1. QED.
I mean the reality is that in the same way that 2 * 0 is ONE.
Or if I were using latex:
2 \times 0 = 1_{+}
Because multiplication, being repeated addition and exponentiation repeated multiplication both behave the same way. When asked to repeat the operation zero times, they return the unit or 1 of the underlying group, which is typically denoted as 1_{whatever}
However, the 1 of addition is 0 when using the standard notation for integers
Declaring 2x2=0+2+2, 2x1=0+2, 2x0=0 while 2xx2=1x2x2, 2xx1=1x2, 2xx0=1 seemed arbitrary.
What helped was learning about negative exponentiation and exponentiation simplification, so 2xx0 = 2xx2 x 2xx(-2) = 2xx2 / 2xx2 = 1.
That said, I clearly still take issue with unintuitive interpretations of "nothing" https://stackoverflow.com/questions/852414/how-to-dynamicall...