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Positive and Negative Exponents

Date: 07/27/97 at 14:26:29
From: Anonymous
Subject: Exponents

Dr. Math,

Could you please explain why (-1)^n = 1 for any even number n, and why 
(-17)^-8 is positive?

Thank you for your help,

Date: 07/28/97 at 13:21:43
From: Doctor Beth
Subject: Re: Exponents

Good question!  The general idea of raising a number to a positive 
integer exponent "n" is to multiply that number by itself n times; for 
example, (-1)^4 = (-1)*(-1)*(-1)*(-1) = 1.  Remember that two 
negatives multiply to be a positive, so that if "n" is even, all the 
negatives can be paired with another negative, and the result is 
positive. So that's why any number to an even positive power is 
positive. (Incidentally, that's why you can't find a real number that 
is the even root of a negative number; for example, there is no real 
number that is the square root of -1, because to be the square root 
of -1, the number squared would have to be -1, which we just decided 
can't happen.)

Now for your second question. Negative exponents are a bit tricky at 
first - they mean that you have to put the number in the denominator 
and take a positive exponent.  In symbols, this is the same as saying 
a^(-b) = -----  .

So (-17)^(-8) = 1/(-17)^8, and since 8 is even, (-17)^8 is positive, 
so that 1/(-17)^8 is positive.

Thanks for the question!

-Doctor Beth,  The Math Forum
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Associated Topics:
Middle School Exponents

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