I would prefer $\frac{1}{x}$. However: for units, instead of $5 \;\text{cm}/\text{s}$ we often see $5 \;\text{cm}\,\text{s}^{-1}$. But that may only be beyond the high school level.

I think that it mostly doesn't matter, but that there are specific circumstances in which one notation might be preferable. For example, the derivative of $x^{-2}$ is more "obvious" to me than the derivative of $\frac{1}{x^2}$. In teaching, I would ensure that students are comfortable with both.

"I know that questions will sometimes request an answer without negative exponents" I think I somehow fail to see the point. Why would one include such a condition in a question?

@JochenGlueck Because students need to practice manipulations. It is about getting students comfortable with the idea that $x^{-3}$ is the same thing as $1/x^3$, and vice versa. You might see similar exercises involving $\sqrt[4]{x}$ and $x^{1/4}$.

@XanderHenderson: Practing manipulations is important, of course. If one tries to achieve it by requiring a specific format for the answer though, chances are that this will just reinforce many students' believe that math were a collection of arbitrary and strange rules which they need to follow to satisfy there teachers. Working with expressions such as $\sqrt[4]{x} x^{1/2}$ or $x^{-2}\cdot 1/x^3$ brings the same practice without artificial constraints on formatting the result, and also reminds the students at the same time why changing from one format to another is useful.

@JochenGlueck I really don't think that you and I are thinking about these exercises in the same way. In a single assignment / quiz / exam, I might ask students to write an expression using only positive exponents, and using exponents with mixed signs, and variations thereof. The exercises are very clearly about manipulating expressions, and not about writing them in some standard form.

@GeraldEdgar For what it's worth, questions on IB math examinations (which students take in high school) tend to express units using negative exponents.

@XanderHenderson: I agree with you that we look at this from quite different perspectives. I'd still be interested in understanding your point better: if an exercise anyway contains such expressions written in different forms and asks the students to simplify, then what is the point of requiring, say, the answer not to contain negative exponents? The exercise will have the students use and rewrite different notations anyway. So why require a certain format of the answer and thus risk to further perpetuate many students' notion that "math is about writing down what the teacher wants to read"?

@JochenGlueck "...if an exercise anyway contains such expressions written in different forms and asks the students to simplify..." The exercise does not contain the phrase "simplify". The exercises I write are generally phrased as "Rewrite the expression so that it contains only positive exponents," or "Rewrite the expression so that it uses positive and negative exponents and has a denominator of $1$" (or something like those). I avoid the instruction "simplify" like the plague, since it is so gosh-darned context dependent.

Once students are comfortable writing expressions in different forms, the next step is to talk about when any particular form is useful. But they have to master the basic manipulations at some point.

@XanderHenderson: Hmm, so if the exercise is "Rewrite $x^2 \cdot x^{-3}$ so that it contains only positive exponents", I assume that "$x^2 / x^3$" would be a valid answer?

@JochenGlueck If that were the instruction, yes. I also usually add an instruction about each base appearing only once in order to avoid this particular edge case, and there are a few other edge cases that students have found over the years that I tend not to try to prevent (I can't think of any off the top of my head...), since the prompt can quickly get out of hand.

E.g. from last spring's precalculus final:

It is technically a college level course, but the students I encounter tend to require a fair amount of remediation. This particular problem is from week 1, and it meant as part of a review of the basic properties of real numbers.

And yes, I fear that US high school teachers tend to focus on computational procedure---exercises like this are first step away from a focus on procedure.

The point made to students is that $1/x$ is precisely the same things as $x^{-1}$ (in the context of real numbers). So you should be able to freely and quickly switch between the two.

In a similar vein, "factor $x^2 + 2x + 1$" vs "expand $(x+3)(x-4)$". Sometimes, we want our polynomials factored. Other times, we would like to see them expanded into "standard form". Or Horner's form. Students should be able to go from one form to another, in order to actually solve whatever problem they are trying to solve.

@XanderHenderson: Thanks for your response! Re "the point made to students is that $1/x$ is precisely the same things as $x^{-1}$": I think I don't see you how your version of the exercise make this point any more than the wording "Simplify the expression $\frac{a^2 b^5 c^4}{a^3 (b^2)^{-3} d^{-4}}$ as much as possible" would (because one still can't solve the exercise without this kind of knowledge).

What does "simplify" mean?

To say "simplify" is to imply that there is one, "most simple" expression, and that the student must determine what that expression is.

To "rewrite" an expression makes no judgement about whether or not the target form is "better" or "worse" than the original form.

@XanderHenderson: I don't see why "simplify" means that there is one "most simple expression". It only means that there is an expression such that there is no strictly simpler expression. There's no uniqueness requirement (which is precisely my point).

@JochenGlueck You are talking about denotation. I am talking about connotation.

Sorry, I don't follow. What do you mean by denotation and connotation in this context?

When you tell a student to "simplify" an expression, they immediately understand that to mean that the given expression is "complicated", and that the goal is to find a "more simple" expression. The best answer is the "most simple" expression.

csun.edu/~bashforth/098_PDF/06Sep15Connotation_Denotation.pdf

"Denotation" is the literal meaning of a phrase. "Connotation" is how it is interpreted by a listener / reader.

Ah, ok, thanks for the link.

When you tell a student to "simplify", they typically understand that to mean "there is a most simple expression; write that expression down".

I'm not so sure whether this is really the first connotation that students have when they here "simplify". They are used to a lot of situations from real life where there is not unique maximum or "best result" - so if we talk about "simplify" and the first thing that students think is that there were a "most simple expression" I think they only do this because they're sitting in a math class. This is yet another instance of my point, students having weird ideas about math.

@JochenGlueck Experience has shown me that, for my students, at least, "simplify" has a very prescriptivist meaning.

And yes, students have weird ideas about math. Best to use language which does not reinforce those ideas.

@JochenGlueck I think that it is bad mathematical style to use the word "simplify". I avoid it in my own writing as much as possible, and encourage others to do the same. I think that it is a vague and ambiguous term.

Again, experience shows me that "simplify" is an instruction which confuses students. It can mean different things in different contexts, and the students that I encounter tend to have a very prescriptivist view of mathematics. Part of my job is to break them of that point of view, and I have found that "simplify" (as an instruction) just makes things harder.

From a pedagogical point of view, I think that it is best to avoid "simplify".

@JochenGlueck I think that it is somewhere in between.

I tend to organize my questions into three broad categories (roughly aligned with Bloom's taxonomy). At the lowest level are questions which are, essentially, memorization tasks, or tasks built on rote computation. For these kinds of problems, I tend to be fairly prescriptive about what answer I am expecting.

These kinds of questions are, in principle, automatable---if I had the energy, I could write them up on, for example, WebWork, and have a computer grade them.

At the middle level, the goal is to "analyze" and "evaluate". More complicated computations might be involved, and answers may not be "unique" (in the sense that a student "simplifying" an expression differently might lead to an answer with a different form). At this level, I don't care what the answer looks like, as long as it is correct.

At the highest level, the problems might be a lot more open ended, and there shouldn't be any one, unique, right answer. Students are encouraged to write their answer in whatever manner they believe most clearly communicates what they've done.

For example, the above is from a written assignment. Question 6 is kind of in that middle zone, while Question 7 is tending toward the highest level.

@XanderHenderson: Ah ok, I think I got a more complete picture of your perspective now. Thanks for the detailed reply and for the interesting discussion!