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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?

```

```
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
that
1
a^(-b) = -----  .
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
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Exponents

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