Combining Positive and Negative Exponents
Date: 06/30/99 at 12:33:38 From: Lauren Fortner Subject: Algebra: using the power theorem I understand how to get problems from "beginning" form, for example: x (x^-3)^2 y (xy^-2)^-3 ----------------------- (y^2)^3 y^-3 (x^2)^3 to "after" you use power theorem: x x^-6 y x^-3 y^6 ----------------- y^6 y^-3 x^6 and then when you simplify the numerator and denominator: x^-8 y^7 -------- y^3 x^6 What I don't understand is how you get from that to writing all exponential expressions with positive exponents, or negative exponents, or with both. I homeschool, so I don't really have anyone to ask. Thank you for your help.
Date: 07/01/99 at 17:00:53 From: Doctor Peterson Subject: Re: Algebra: using the power theorem Hi, Lauren. Thanks for a well-written question - it really helps to know just what part you have trouble with. The key to this step is that x^-a 1 1 x^a ---- = --- and ---- = --- 1 x^a x^-a 1 That's essentially just the definition of negative exponents, and I'll assume you're at least aware of this as a fact. What you need is how to apply it. What these facts mean in practice is that you can move a factor from top to bottom or from bottom to top and negate the exponent. Once you get used to it, you just think "There's an x^-8 in the numerator, so I can replace that with an x^8 in the denominator." To take it more slowly, we can pull the expression apart, apply the rule, and put it back together: x^-8 y^7 x^-8 y^7 1 1 1 y^7 1 1 -------- = ---- * --- * --- * --- = --- * --- * --- * --- y^3 x^6 1 1 y^3 x^6 x^8 1 y^3 x^6 y^7 = ----------- x^8 y^3 x^6 That makes all the exponents positive, but there's still another step you didn't mention: combining like factors so that x and y each appear only once. That's easy in the denominator now; we can just permute so the x's are together and add the exponents: y^7 y^7 ----------- = -------- x^8 y^3 x^6 x^14 y^3 But you still have y in two places. We can use the same rule to get the y's together; since 7 > 3, let's move the y^3 to the top to keep the exponents positive: y^7 y^7 1 1 y^7 1 y^-3 y^7 y^-3 y^4 -------- = --- * ---- * --- = --- * ---- * ---- = -------- = ---- x^14 y^3 1 x^14 y^3 1 x^14 1 x^14 x^14 Now we're really done. But we've taken a lot more steps than we had to. That's fine when you're starting out; this isn't a race. But here's how I'd do it myself: x(x^-3)^2y(xy^-2)^-3 x x^-6 y x^-3 y^6 -------------------- = ----------------- (y^2)^3y^-3(x^2)^3 y^6 y^-3 x^6 = x x^-6 y x^-3 y^6 * y^-6 y^3 x^-6 (Here I've moved everything to the top.) = x x^-6 x^-3 x^-6 * y y^6 y^-6 y^3 (Here I've gathered the x's and y's together; I'd probably mark each factor as I copied it to make sure I didn't miss any.) = x^(1-6-3-6) y^(1+6-6+3) = x^-14 y^4 y^4 = ---- x^14 If you're at all afraid of negative exponents, this will make you dizzy, but if you like to take the bull by the horns and get it over with, making the negative exponents work for you, this is the way to do it. Basically, I decided I'd rather deal with positive and negative exponents than with numerators and denominators; they're really two ways to say the same thing, and mixing them doesn't make sense. So even though the goal is to have numerator and denominator and eliminate negative exponents, I found that it's really easier to work with the negatives, then change them to denominators when I'm done. If any of this overwhelmed you, or if you have more questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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