Expert: Sherry Wallin - 12/2/2009

Question
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: Ben, one way to think of this is this: let's take a simple example in base 10:
10^1 = 10, 10^2 = 10*10 = 100, 10^3 = 10*10*10 = 1000...
what about 10^-1? = 1/10, 10^-2 = (1/10)(1/10) = 1/100, 10^-3 = (1/10)(1/10)(1/10)...
you need something to take you from 10^-1 to 10^1 and that just happens to be 10^0. So between 1/10 and 10 you have the number 1. Another way to think about this is if you have 10^-1 and you multiply it by 10^1 using laws of exponents you have 10^(-1+1) = 10^0 but you know that
10^-1 * 10^1 = 1/10 * 10 = 1, don't you?? So it must be that 10^0 is 1.

Math Prof

---------- FOLLOW-UP ----------

QUESTION: laws of exponents? what are those?

Answer
When the bases are the same we add exponents, example: 2^2 * 2^3 = 2^(2+3) = 2^5 = 32 you can check this by calculating 2^2 = 4 and 2^3 = 8 and 4*8 = 32 but when you have variables you can't always check it so you need rules. When the bases (and in the example 2 is the base and the exponents were 2 and 3 respectively) are the same add the exponents. When you have a power to a power as in (2^2)^4 = 4^4 = 2^(2*4) = 256 then you multiply the exponents.

Math Prof