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I see all the time in high school math textbooks problems saying to "simplify" an expression. Their explanations of what it means for an expression to be "simplified" is a bit vague and does not reflect the rigor mathematics usually provides:

"To simplify an expression means to do all the math possible." (OpenStax Intermediate Algebra)

I have encountered numerous situations in which there is ambiguity as to what looks "simpler" to people. For example: $$ \text{Simplify: }\frac{a^2b^{-3}c}{c^2b} $$

Some people would say the most "simple" way to write this is $a^2b^{-4}c^{-1}$. However, most textbooks would consider $\frac{a^2}{cb^4}$ to be the "correct" answer.

From my perspective, problems beginning with "Simplify" don't really make sense (and are frankly unfair to the students) unless there is a clear definition of what it means for the expression to be "simplified" (especially with teachers who are picky about that kind of thing).

Is there some sort of convention for simplification that I don't know about that makes this work?

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    $\begingroup$ I doubt it. From what I've observed while in school, "simplest" usually just means "cleanest expression", which is highly subjective. Excellent question! $\endgroup$ – Alann Rosas Nov 9 '20 at 22:20
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    $\begingroup$ Both $a^2b^{-4}c^{-1}$ and $\frac{a^2}{cb^4}$ should count as simplified and as correct $\endgroup$ – Henry Nov 9 '20 at 22:20
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    $\begingroup$ @SenZen well, with proofs and other applications of Algebra, it's more about manipulating the expression to meet your purposes, rather than a generic "simplification" $\endgroup$ – Caleb H. Nov 9 '20 at 23:30
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    $\begingroup$ @SenZen the same way you learn the generic simplifications - the textbook/teacher generally gives examples of how to use the different rules (i.e. the rule of negative powers) in the context of "simplifying". Why not just teach it in the context of an equation instead, where there is a purpose to the simplification? $\endgroup$ – Caleb H. Nov 10 '20 at 17:26
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    $\begingroup$ @SenZen I think you're misunderstanding me. I certainly think the rules for manipulating the expressions (in your case, division) need to be learned before they can be properly used. I just don't like the specific wording "simplify" because it is rather subjective. $\endgroup$ – Caleb H. Nov 10 '20 at 23:38
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This is right:

From my perspective, problems beginning with "Simplify" don't really make sense (and are frankly unfair to the students) unless there is a clear definition of what it means for the expression to be "simplified" (especially with teachers who are picky about that kind of thing).

In K-12 "simplify" usually means "manipulate so that you end up with an expression of the form we've seen as the answer to similar problems".

That is indeed ambiguous. It does however give students practice with the kinds of manipulations they are supposed to learn how to do. It makes exams easier to grade. How much it helps students learn interesting and important mathematics is doubtful. Good students understand the ambiguity, but still do what the teacher expects. If they have good teachers they can discuss the ambiguity in a way that does not disturb their classmates who just want to get the right answer.

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    $\begingroup$ Right you are. For my money, the only proper use of the word is in the sentence, “Don’t simplify your answer.” $\endgroup$ – Lubin Nov 10 '20 at 2:49
  • $\begingroup$ "How much it helps students learn interesting and important mathematics is doubtful" You're kidding right? Good luck learning anything beyond natural number arithmetic without learning generic simplifications first... $\endgroup$ – user838035 Nov 10 '20 at 23:30
  • $\begingroup$ And as for the "ambiguity is unfair" perspective, well, would you suggest we only teach math with clear precise definitions? Sorry kiddo, I know you haven't learnt addition yet, but the peano axioms come first $\endgroup$ – user838035 Nov 10 '20 at 23:35
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    $\begingroup$ @SenZen You do need to become proficient at algebraic manipulation, just as you need to learn the multiplication tables. But the manipulations, like the tables, are a tool, not an end in themselves. $\endgroup$ – Ethan Bolker Nov 11 '20 at 0:51
  • $\begingroup$ @Ethan Bolker Everything is tool, and everything is an end in itself, depending on what you're trying to achieve. $\endgroup$ – user838035 Nov 11 '20 at 8:24

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