First, for the myth debunking, a/k/a “When a student tells you one thing and your teacher tells you a different thing, which one is more likely to be right?’
We worked on this problem in class:
$\left(\frac{5a^{4}z^{−7}}{3a^{−6}z{−4}}\right)^{−3}$
The student who was working it at the board started out by trying to simplify what is inside the parentheses first, which, I would like to point out, is absolutely correct: however, she made a mistake in that simplification. When I stopped her and said there was an error, several students told her that she was “supposed” to change the -3 power into the 3rd power of the reciprocal. (Not necessarily in those words.) I then countered by saying as insistently as I could that no, she did not have to do that first, she could simplify inside the parentheses first, I was only objecting that she had made an error in simplifying.
So what happened? The student listened to the students, who were WRONG in what they said.
It would have been totally correct to simplify inside the parentheses first and then deal with the -3 power. As I will do here:
$\left(\frac{5a^{4}z^{−7}}{3a^{−6}z{−4}}\right)^{−3} = \left(\frac{5a^{4}a^{6}z^{4}}{3z{7}}\right)^{−3}$
$= \left(\frac{5a^{10}z^{4}}{3z{7}}\right)^{−3}$
$= \left(\frac{5a^{10}}{3z{3}}\right)^{−3}$
$= \left(\frac{3z{3}}{5a^{10}}\right)^{3}$
$= \frac{27z{9}}{125a^{30}}$
Morals of the story: there are at least two!
• When a student says one thing and the teacher says the opposite, maybe it’s better to listen to the teacher?
• There is not only one correct way to work these problems, and, indeed, that is true of a lot of what we do in this course. The important thing is to know the correct properties and definitions and how to use them correctly: they are your tools. And then, in working a problem, you look at what you have at each step and ask, what can I do with this? What tools can I use that may get me where I want to go? As long as you use mathematically correct “tools” and use them correctly and get to what the problem was asking for, your work is correct.
Thus it is your job to look to the examples to understand why we choose to do what we do at each step: what in the problem itself is suggesting to use this particular tool? Avoid at all costs memorizing “steps” whose purpose you do not understand. That is the opposite of mathematics!
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