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Rule collision

The same experience has happened to me several times in the Maths Drop-In Centre recently — with different students from different courses — and it was such a strong pattern I need to talk about it.

The students are doing some algebra involving negative powers on the tops of fractions.  Something like this:

(1-x^(-2))/(1+x^(-2))

Now they remember this rule (probably from school) which says that a negative power belongs on the bottom of a fraction but as a positive power. And so they do one of these:

(1-x^2)/(1+x^2) or (1+x^2)/(1-x^2)

 

Both of thes are, of course, TOTALLY WRONG. But the students have a hard time being convinced of this fact.

The problem is, that that rule only works if everything involved in your fraction is multiplication and division. It doesn’t interact with the plus and minus that are trapped there on top and bottom of the fraction. And why doesn’t it interact with the plus and minus? Because the rule is based on the definition of what a negative power means. This is what it means:

x^{-2}=1/x^2

What this means is that multiplying by a negative power is the same as dividing by the matching positive power. And this gets to the heart of the issue: adding a negative power is not at all anything to do with multiplying it, so the nice “switch to the bottom, make positive” rule just isn’t going to work, because you have to do the addition first.

The rules for negative powers are colliding with the rules for addition, and for fractions, with unpredictable results! If only the students had been encouraged more to work from the original definition rather than it being all about remembering a rule. Then maybe the results wouldn’t be quite so unpredictable! If only the students had attempted a few things like this in the past in a situation where someone could notice it and talk to them about it! Then maybe they would have found this glaring gap in their understanding of algebra!

PS: If you’re wondering how to go about simplifying that fraction, then you have to deal with the negative power using its original definition — which means it will become a positive power on the bottom of its very own little fraction first. Like this:

eventually comes to (x^2-1)(x^2+1)

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