Positive and Negative ExponentsDate: 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, BradyRed 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/ |
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