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Numbers Raised to the Negative Power

Date: 11/14/2001 at 10:35:33
From: Rosemary
Subject: Numbers raised to the negative power

I am in 7th grade and in Pre-Algebra. We are using numbers raised to 
the negative power. I know that 5^(-N) = 1/5^N. I would like to know 
why. My teacher said it was by definition, but I know there is a 
reason behind this definition. Any help would be appreciated.  

Thanks in advance.  

Date: 11/14/2001 at 12:18:53
From: Doctor Peterson
Subject: Re: Numbers raised to the negative power

Hi, Rosemary.

Your teacher is right that negative exponents are defined this way; 
but the definition is made that way for a good reason, namely so that 
the properties of exponents continue to be true.

Look at this expression:

    a^b * a^-b

If the property

    a^b * a^c = a^(b+c)

still holds for negative exponents, then we find that

    a^b * a^-b = a^(b + -b) = a^0

Therefore, our definition has to say that

    a^-b = a^0 / a^b

Now, how do we define a^0? Again, consider

    a^1 * a^0 = a^(1+0) = a^1

This means that

    a^0 = a^1 / a^1 = 1

Now we know that we have to define a^0 as 1, and that we have to 
define a^-b as 1/a^b.

See the Dr. Math FAQ for more ideas in this area (mostly about a^0):

   n^0 (any number to zero power)   

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Exponents
High School Number Theory

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