Negative Exponents ExplainedDate: 12/29/2001 at 16:57:12 From: Alec Subject: Solving for x The question is, 5 to the negative second. I have a slight problem with exponents and I was hoping you could shed some light on the subject Thanks. Date: 12/29/2001 at 19:00:13 From: Doctor Achilles Subject: Re: Solving for x Hi Alec, Thanks for writing to Dr. Math. First, a bit about notation. I use the ^ symbol to designate exponents. So 3^4 is "three to the fourth" or a 3 with a little 4 above and to the right. And 2^(-5) is "two to the minus fifth" or "two to the negative 5th." Negative exponents are something that I think is not taught very well in most schools. Here's what most people are taught: A negative exponent means you just turn the number into a fraction and then take the positive exponent. So 3^(-2) is the same thing as (1/3)^2 or (1/3)*(1/3) or 1/9. And 5^(-3) is the same thing as (1/5)^3 or (1/5)*(1/5)*(1/5) or 1/125. This is correct, but it doesn't really explain _why_ it is correct. Here's how I like to think about negative exponents. I start by thinking about positive exponents. Let's look at powers of 5: 5^3 = 5*5*5 = 125 5^2 = 5*5 = 25 5^1 = 5 = 5 Now, how do I get from 5^3 to 5^2? Here's another way to look at it: 5^3 = 125 5^2 = 5^3/5 = 125/5 = 25 So 5^2 is really just 5^3 divided by 5. 5^1 = 5^2/5 = 25/5 = 5 And 5^1 is really just 5^2 divided by 5. and we can keep going: 5^0 = 5^1/5 = 5/5 = 1 Aha! So _that's_ why people always say "anything to the zero power is 1." 5^(-1) = 5^0/5 = 1/5 So 5^(-1) is really just 5^0 divided by 5. 5^(-2) = 5^-1/5 = ? I'll let you finish that off. All you have to remember is that 5^(-2) is really just 5^(-1) divided by 5. Hope this helps. If you have other questions about this or you're still stuck, please write back. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ |
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