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Zero as an Exponent


Date: 7/15/96 at 16:40:9
From: Anonymous
Subject: Exponent of 0

Why do you get 1 when you take a number to the 0 power?

Thanks!

Thai-Meister


Date: 7/15/96 at 17:8:27
From: Doctor Paul
Subject: Re: Exponent of 0

Let's first look at an example.  Let's look at the list of numbers
      3^1, 3^2, 3^3, 3^4, ....   Finding the actual values, we get
        3,   9,  27,  81, ....

So what is the pattern in the bottom sequence? Well, every time you 
move to the right in the list you multiply by 3, and every time you 
move to the left in the list you divide by 3. So we could take the 
bottom sequence and keep going to the left and dividing by 3, and we'd 
have the sequence that looks like this:

..., 3^-3, 3^-2, 3^-1, 3^0, 3^1, 3^2, 3^3, 3^4, ....     
..., 1/27,  1/9,  1/3,  1,   3,   9,   27,  81, ....

So now we know what all the powers of 3 are! Actually, we just did 
the integer powers of 3. But that's probably enough for now.

Now the trick is to show that you can do the same thing for any 
number, not just 3. It's actually not all that hard. So once you do 
that, you'll see that the x^0 term is always going to be 1, so that 
means that x^0 is 1 for all x (except maybe 0^0, but that's a whole 
other discussion)!

-Doctor Paul,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Exponents
High School Number Theory
Middle School Exponents

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