Zero as an Exponent
Date: 2/1/96 at 19:11:57 From: Anonymous Subject: powers of zero Dear Dr. Math, We know that 2 to the power of zero is one, but we have a problem understanding some of the reasoning. Two to the power of 3 is 2 times itself 3 times or 2x2x2. But, two to the power of 0 isn't 2 times itself 0 times, is it? Can you explain 2^0 in those terms? Thanks, Kathryn Gerleman Jordan Middle School 7th grade pre algebra
Date: 3/12/96 at 1:24:37 From: Doctor Jodi Subject: Re: powers of zero Hi Katryn! This is a really good question! I'm not sure if you're familiar with negative exponents, and they always confuse me a bit, so let me start off with Ken's explanation of negative exponents. Negative exponents mean that instead of multiplying that many of the base together, you divide. For instance, 3^2 = 9, and 3^-2 = 1/9. That's one way to see why anything to the zero power (except perhaps 0) is 1. The way most people think of negative exponents is "put it in the bottom of the fraction. So, let's look at a series: 3^1 = 3 3^2 = 9 3^3 = 27, etc. On the other side, we can add some negative exponents: 3^(-1) = 1/3 3^(-2) = 1/9 3^(-3) = 1/27, etc. So we have two series that look like ... 1/27, 1/9, 1/3 and 3, 9, 27.... (The dots mean that you could continue the series in that direction if you wanted to ... for as long as you wanted.) Do you see any pattern? Well, in each series, each time you jump to the right, you multiply by 3. But do you also notice that 1/3 and 3 have a relationship of multiplying/dividing by 9 (depending which way you're going)? Well, if we add 1 as a sort of mathematical glue between the two series, we get ... 1/27, 1/9, 1/3, 1, 3, 9, 27 Which I think is much prettier than the two series above. In order to make this series, we need 1... which corresponds to 3^0. There are other explanations, too, that deal more with adding and subtracting exponents. If you want to know more, or if you have other questions, write us back! -Doctor Jodi, The Math Forum
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