```Date: Oct 2, 2012 4:43 PM
Author: Dave L. Renfro
Subject: Re: How to simplify an expression ...

DCJLEE@AOL.COM wrote:http://mathforum.org/kb/message.jspa?messageID=7899515> I've noticed through the years that many students had> considerable difficulty simplifying the expression below > on a test in Intermediate Algebra:> > [ (6 r^8 t) /(-3 r^2) ]^3> > I'm interested in seeing how you think students should > approach this problem, and detailed steps they should > take, along with precise assumptions/formulas they> should know and be able to apply correctly at each step.> Thank you in advance.I would tell them that if a fraction anywhere in sight(that isn't being added to or subtracted from another fraction)can clearly be reduced, then they should first reduce thefraction. This assumes, of course, that students canrecognize when a fraction can be easily reduced and thatthey can easily carry out the reduction of the fraction.If this is a problem for students, then I would focuson working on just that aspect, and not getting intosomething like the above.Doing this leads us to [-2*r^6*t]^3, which is just[-2 * r^6 * t] * [-2 * r^6 * t] * [-2 * r^6 * t]= [-2 * -2 * -2] * [r^6 * r^6 * r^6] * [t*t*t]= (-2)^3 * (r^6)^3 * t^3= -8 * r^18 * t^3.I've found that it helps if you often fill in the step ofreplacing an exponent with repeated multiplication,rather than simply using a rule that some students may stillbe unsure of. In the calculation above you can then go backand show how our work essentially verifies the identity(ABC)^3 = A^3 * B^3 * C^3. Certainly do this when studentsare still learning and practicing properties of exponents,although I've even found it helpful in calculus coursesfrom time to time. Also, I might (on a blackboard) usea different color of chalk and, off to the side with anarrow drawn to the appropriate place, write-and-box somethinglike this: r^6 * r^6 * r^6 = (rrrrrr) * (rrrrrr) * (rrrrrr),which is a total of 18 r's being multiplied.As for reducing fractions, here's an approach I often usewhen explaining the process to students.Break up the fraction into separate fractions beingmultiplied (remind students how one multiplies fractions)so that like things are with like things. I'm reminded ofthe cafeteria scene in the 2004 movie "Mean Girls" where oneperson says: "Where you sit in the cafeteria is crucial'cause you got everybody there. You've got your Freshmen,ROTC guys, Preps, JV Jocks, Asian Nerds, Cool Asians, VarsityJocks, Unfriendly Black Hotties, Girls Who Eat Their Feelings,Girls Who Don't Eat Anything, Desperate Wannabes, Burnouts,Sexually Active Band Geeks, The Greatest People You Will EverMeet, and The Worst. Beware of the Plastics." However, it'sprobably not a good idea to bring up this in an actual mathclassrooom (or you'll find that, among the many later retellingsof this by students to their parents and friends, enough ofthe original context will be lost in a few cases that you'llbe accused of advocating racism), but one-on-one tutoringmight be O-K.Doing this to (6 r^8 t) / (-3 r^2) leads to(6 / -3) * (r^8 / r^2) * (t / 1)Of course, the fractions would be written in vertical form.Now reduce each separate fraction:- -(2 / 1) * (r^6 / 1) * (t / 1)= (-2 * r^6 * t) / 1*1*1= -8 * r^6 * tFinally, I might (on a blackboard) use a different color ofchalk and, off to the side with an arrow drawn to the appropriateplace, write-and-box something like this:r^8 / r^2= (rrrrrrrr) / (rr)= (r/r) * (r/r) * (r/r) * (r/r) * (r/r) * (r/r) * (r/1) * (r/1)= 1*1*1*1*1*1*r*r= r^2One of the things I'd hope to get across with this extrawork is that you can often do these calculations even ifyou've forgotten some exponent rules. Just write down whateverything is without using exponents. Also, it shows thatthese exponent rules aren't arbitrary "math grammar" rules,but rather these rules simply represent shortcuts one cannotice, at least in the case for positive integer exponents.Dave L. Renfro
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