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Most problems require the student to simplify the final answer and in many context the meaning is obvious: For example, $3/9$ should be simplified to $1/3$ and $\sqrt{16}$ should be simplified to $4$.

However, how about the answer $$6\left(\frac{2}{3}\right)^{1/5}?$$

This same answer can be equivalently written as $\sqrt[5]{5184}$ or $2^{6/5}\cdot 3^{4/5}$, etc. Which one(s), if any, is considered as "simplified as much as possible"?

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    $\begingroup$ There very rarely is such a thing widely agreed upon as a canonical form for answers; students should be allowed to use whatever is convenient for their intended purpose at hand. $\endgroup$ – Vandermonde Jan 31 at 5:53
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    $\begingroup$ My beef is that simplify usually means: For this test, and for this question on the test, I know what "simplify" means. You are to use this meaning. ["I" does not refer to myself.] In my case, I always tried to be explicit with how I wanted the final answers to appear, and for something like what you have, it just wouldn't come up, because if this was the answer to something else, then the "something else" is what grading would be based on, and if the task was to rewrite this into some other form, then I'd pose a problem that would presumably require the rewrite to solve. $\endgroup$ – Dave L Renfro Jan 31 at 13:20
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    $\begingroup$ As others have said, it depends on what you want to do with the number. Are we comparing it to another 5th root? Are we deciding if it is irrational? Are we going to plug it into a polynomial with a lot of $\frac{1}{6}$'s? My CAS tells me $2\cdot 3^{4/5} \cdot 2^{1/5}$ is simplest. Free from any other context, I usually go for "uses the minimum number of operations" as a definition of simplest. $\endgroup$ – Adam Jan 31 at 16:51
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    $\begingroup$ @Zuriel Ok, but what does the number mean? $\endgroup$ – Adam Jan 31 at 16:59
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    $\begingroup$ @Zuriel For example, in applications, numbers indicate things about physical systems and often have units. In number theory, the number might indicate something about the factors of another number. It's OK if it doesn't actually mean something and the students were just practicing some formal manipulation, but if there were some purpose to which they were going to put the number to, that would likely decide what "simplest" means. $\endgroup$ – Adam Jan 31 at 18:55
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Rigid criteria for simplification seem to me largely a bad idea if they are not motivated by contextual considerations. The idea that $\sqrt{2}/2$ should be preferred to $1/\sqrt{2}$ struck me as unmotivated when I was a student, and now seems to me problematic to motivate. The situation is different with respect to writing rational numbers or rational functions. In both cases there is a canonical simplification, that in which the numerator and denominator are relatively prime.

In a particular context it might make more sense to write $6(2/3)^{1/5}$ than to write $2^{6/5}3^{-4/5}$. It might be that the $6$ in front and the $2/3$ have different extra-mathematical origins, e.g. the $6$ counts something and the $2/3$ reflects a ratio of measurements, and it's just an accident that both are expressible in terms of powers of $2$ and $3$. In a purely mathematical context there are no such considerations, but then still one might prefer one form or the other (I find $6(2/3)^{1/5}$ easier to interpret at a glance than $2^{6/5}3^{-4/5}$).

Returning to the $\sqrt{2}/2$ versus $1/\sqrt{2}$ example, imagine we are working in the field $\mathbb{Q}[\sqrt{2}]$. Then, in fact, we can always write an element of $\mathbb{Q}[\sqrt{2}]$ in the form $a + b\sqrt{2}$ with $a$ and $b$ rational. In this context there is a normalized expression, and it would be $(1/2)\sqrt{2} = \sqrt{2}/2$ rather than $1/\sqrt{2}$ (working in $\mathbb{Q}[\sqrt{2}]$ we have to prove (!) that $\sqrt{2}$ is invertible before we can give sense to $1/\sqrt{2}$ - as a shorthand for its inverse in this field extension of $\mathbb{Q}$). On the other hand, if we are working in the field $\mathbb{R}$, then $1/\sqrt{2}$ is directly the expression for the inverse of $\sqrt{2}$ qua element of $\mathbb{R}$. It's that thing that gives $1$ when we multiply it by $\sqrt{2}$ which we know exists from when we constructed the real field (when did we do that?). The conceptual apparatus necessary to make such distinctions/motivations comprehensible (in this case amounting to distinguishing between the inverse of $\sqrt{2}$ in the field $\mathbb{R}$ and its inverse in $\mathbb{Q}[\sqrt{2}])$) is simply not available to (all but a very few highly exceptional) students in primary education where these things are taught. Insisting on the use of a "canonical" form without providing motivation for what makes it canonical ("canonical" is always with respect to some background context) has the feel of authoritarian dogma. This can alienate students, particularly those with a more creative or curious mindset.

Absent some context that motivates preferring one or another simplified form, demanding one or the other such form seems to me typical of the sort of teaching that turns students off from mathematics. What is arbitrary should generally be avoided.

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    $\begingroup$ While working on an example in an undergrad precalculus course, I found the answer to be $1/\sqrt{2}$ and left it there. A student said, "I was told we are not allowed to leave an answer like that." I said, "That is what you were told in high school. Here's why ..." and went through a short historical tangent about arithmetic in the pre-computer days, tables of square roots, and long division. I thought I explained the issued well. On end of semester evaluations, I got a comment: "Teacher said our high school math education was wrong." $\endgroup$ – Brendan W. Sullivan Jan 31 at 16:54
  • $\begingroup$ omg, this made me laugh. It's not just high school. It's other math teachers. Students' attention (almost by necessity) fades in and out, so they will take odd meanings from what we say. I just gave this explanation (short version) today. Before calculators... I just showed how I could do it in my head if the square root of 2 was in the numerator, and said that mattered before calculators and doesn't now. $\endgroup$ – Sue VanHattum Feb 5 at 19:46
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I would say the canonical answer for what constitutes 'simplified as much as possible' is whatever the exam board says it is.

'Simplify' isn't a mathematical function. It is a pedagogical instruction trying to require students to make use of a selection of mathematical equivalences that they are expected to know. However, it is too vague a term to make clear exactly which such equivalences are expected in a particular case.

Consider

$$\frac{(x-1)(x-2)^2}{(x-2)(x+2)}.$$

A student might reasonably realise they should cancel the factor on the top with that on the bottom, when it's written like this. But if it was presented as

$$\frac{x^3-5x^2+8x-4}{x^2-4}$$

it is far less clear that cancelling can reasonably be expected. Also, whether the final answer should be factorised or expanded is dependent on context.

So I would say that the instruction to 'simplify as much as possible' should only be used in situations where it is clear to the students what that means. Otherwise, more explicit instructions should be used (or more than one answer judged as correct).

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There is a theorem which says that it is impossible to decide the equivalence of two elementary functions syntactically. So there is not, and cannot, be a uniquely defined "simplest form" for a given expression.

http://inst.cs.berkeley.edu/~cs282/sp02/readings/caviness.pdf

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  • $\begingroup$ Thanks! So the requirement "simplify your answer" should be abandoned? $\endgroup$ – Zuriel Jan 31 at 14:49
  • $\begingroup$ @Zuriel I would say so. I never ask for "simplification". Students should, of course, be able to manipulate an algebraic expression into equivalent form in ways that are useful for their problem. $\endgroup$ – Steven Gubkin Jan 31 at 15:04
  • $\begingroup$ So for the final answer, $\frac{8}{2}$ is as good as $\sqrt{16}$ which is as good as $3.9999\cdots$? $\endgroup$ – Zuriel Jan 31 at 15:15
  • $\begingroup$ That is what I do. Now if the question asked something like "Is this a better deal than getting it for $5?", then maybe it would be more convenient for the student to perform some calculation which allowed them to make a comparison." $\endgroup$ – Steven Gubkin Jan 31 at 15:24
  • $\begingroup$ Could you provide a link/reference to this theorem? $\endgroup$ – Daniel R. Collins Jan 31 at 16:05
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I suggest to do a Google search and at least read a couple pages that give the classical rules for simplifying radical expressions. You may still have some critique but will at least know what you are criticizing. Question as stated shows lack of familiarity with basics.

P.s. $2 \sqrt[5]{162}$ is the classical answer.

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    $\begingroup$ Part of the description of the downvote button for questions: "does not show any research effort". Anyways, the answer would be improved by explaining the reasoning behind the classical answer. $\endgroup$ – Tommi Brander Jan 31 at 6:18
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    $\begingroup$ Yeah I could improve it that way. But honestly that would be too much "peel me a grape". It is one thing to ask a question about a concept or to crticize it or not to understand it. But this question shows the lack of even a 1 minute Google search. And it is coming from an experienced teacher. He can do better. Just like the "homework questions" at MSE should show some evidence of effort and initial approach to garner help. $\endgroup$ – guest Jan 31 at 6:21
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    $\begingroup$ I would recommend downvoting the question (if you had the reputation) and moving on, if one is not willing to write a good answer. Plus a comment on how to improve the question, if one is feeling nice. $\endgroup$ – Tommi Brander Jan 31 at 6:42
  • $\begingroup$ @Guest, Could not agree more, but as you can see you got downvoted, and it just rubs me the wrong way. I see such things more and more, esp. on other SE sites. I mean "serial" downvotes just because you didn't explain things or somebody didn't like your answer. "You should've peeled the grape" and "you don't understand" –you can hear in reply. This is wrong. +1 but folks hardly care for new users, new teachers, or new students. That said, there are English mistakes in your post and nobody corrected them. I'm not correcting them either after reading too many posts on Meta SE. That's wrong too. $\endgroup$ – Ken Draco Feb 2 at 4:21
  • $\begingroup$ @Guest, I'm sorry you got such feedback. Actually, Math SE is a very good community. A lot of great answers are here, and a lot of knowledgeable folks are here. But sometimes things may go wrong. $\endgroup$ – Ken Draco Feb 2 at 4:29

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