“Any number to the zero power is equal to 1,” my teachers would say. “Why?” the students would ask. “Because.” the teachers would declare. And this would usually end it. Sometimes a persistent student would again ask, “Why?” to which he’d get the slightly more creative answer, “That’s just the way it is.” Because of the mystery surrounding the zero power as a kid, I always got the feeling that there was something supernatural about it; my teachers held the key and didn’t want to share.
It wasn’t until graduate school that I finally found out why a number to the zero power always equaled one and how to explain this once mysterious phenomenon.
Now as a math teacher, I take the time to explain the zero power. It’s not mysterious, it’s just division. Whenever a question about the zero power comes up, I stop, go to a side board, and ask “What’s 2^3? How about 2^2?” When the kids answer “8” and “4”, I then ask, “Ok, what’s “2^1?” I create a table of their answers on the board:
2^3 2^2 2^1
8 4 2
We then step back and look at the pattern.
Soon the students see that the common difference between numbers is 2. “We divide by 2 to get to the next number,” they say. I then go on to ask, “Then what’s 2^0?”
Of course most students will answer 0 at first. I’ve realized that it’s a natural reaction to answer “zero” whenever hearing “zero” and “multiply” or “divide” within the same lesson. Other students will put 2^0 into the calculator and answer “1”, but that’s what I did as a student and it wasn’t good enough. I always wanted to know why.
I direct all students back to the table we created to find the next term “2^0”.
2^3 2^2 2^1 2^0
8 4 2 ??
“If we divide by 2 each time to get to the next term, what’s 2 divided by 2?” I ask. The students will answer, with an “oh, obviously!” tone, “1.”
Another way to think of this concept is through exponent rules:
To divide terms with exponets, we subtract the exponents, here giving us a 0 exponent. If we then back up and look at the original fraction, we have a number over itself, which is equal to 1. Therefore, any number to the 0 power is equal to 1.