> 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 the fraction. This assumes, of course, that students can recognize when a fraction can be easily reduced and that they can easily carry out the reduction of the fraction. If this is a problem for students, then I would focus on working on just that aspect, and not getting into something 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 of replacing an exponent with repeated multiplication, rather than simply using a rule that some students may still be unsure of. In the calculation above you can then go back and show how our work essentially verifies the identity (ABC)^3 = A^3 * B^3 * C^3. Certainly do this when students are still learning and practicing properties of exponents, although I've even found it helpful in calculus courses from time to time. Also, I might (on a blackboard) use a different color of chalk and, off to the side with an arrow drawn to the appropriate place, write-and-box something like 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 use when explaining the process to students.
Break up the fraction into separate fractions being multiplied (remind students how one multiplies fractions) so that like things are with like things. I'm reminded of the cafeteria scene in the 2004 movie "Mean Girls" where one person 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, Varsity Jocks, 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 Ever Meet, and The Worst. Beware of the Plastics." However, it's probably not a good idea to bring up this in an actual math classrooom (or you'll find that, among the many later retellings of this by students to their parents and friends, enough of the original context will be lost in a few cases that you'll be accused of advocating racism), but one-on-one tutoring might 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 * t
Finally, I might (on a blackboard) use a different color of chalk and, off to the side with an arrow drawn to the appropriate place, write-and-box something like this:
One of the things I'd hope to get across with this extra work is that you can often do these calculations even if you've forgotten some exponent rules. Just write down what everything is without using exponents. Also, it shows that these exponent rules aren't arbitrary "math grammar" rules, but rather these rules simply represent shortcuts one can notice, at least in the case for positive integer exponents.