Expert: Socrates - 12/1/2009

Question
in school i am ahead in my math class so the teacher will usually give me more advanced work on the side. at one point he asked me how would/could i prove that 0^0=1... due to confusion in this problem i put 0^0 in my calculator. it said ERROR. this happened on all calculators. the teacher told me that anything to the power of zero equals one. so i turn to my calculator and type 1^0, 2^0, 3^0, 4^0, 5^0 and so on for the rest of the class!


my question to you is how does ANYTHING(forget about zero), ANYTHING to the power of zero equal one?
i mean if you have 8 sets of 0s, that equals zero!
if you have 8_0s then you still have zero!
if you 0_8s then you still have ZERO
if you have 10^0 IT SHOULD BE 0000000000... 10_0s!!! not 1!
help me please!

Answer
0^0 is not defined to be 1 , there is no good way to define 0^0 . Mathematicians leave it undefined.

There are reasons why it might seem reasonable to set 0^0 = 1 , for example , if you use your calculator you find

.001^.001 = .993

.00001^.00001 = .9999

.0000001^.0000001 = .99999

So it looks like 0^0 should be 1 , because as the base and the exponent get smaller , the entire expression gets closer and closer to 1 . It looks like this shows that 0^0 should be 1. Maybe this is what your teacher had in mind when he asked you this question.

Unfortunately , as you have noted, 0 raised to any positive exponent, no matter how small, is just 0, so this might lead you to define 0^0 = 0 , as you want to do.

So there are really two reasonable ways to define 0^0 , and we can't have it both ways, so it is just not defined at all.