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Say there's an exam question whose answer is $x$ to the power of negative one. Two ways of writing this are $x^{-1}$ and $\frac{1}{x}$.

I know that questions will sometimes request an answer without negative exponents, but if no such directions are given, is one notation preferable to the other?

Context: I'm a middle and high school mathematics teacher and follow the International Baccalaureate system.

Edit: Just to be clear, I'm not asking about marking anything correct or incorrect, or simplifying, or trying to fuss over something insignificant (or worse, encourage students to do so). I was just wondering if there was a reason to prefer one form over another. Seems that it can be context dependent (calculus, trigonometry, units, etc.) but in general, one form isn't to be preferred over the other.

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  • $\begingroup$ 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. $\endgroup$ Commented Aug 19, 2023 at 12:51
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    $\begingroup$ 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. $\endgroup$
    – Xander Henderson
    Commented Aug 19, 2023 at 14:36
  • $\begingroup$ @GeraldEdgar For what it's worth, questions on IB math examinations (which students take in high school) tend to express units using negative exponents. $\endgroup$ Commented Aug 20, 2023 at 13:03
  • $\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Mathematics Educators Meta, or in Mathematics Educators Chat. Comments continuing discussion may be removed. $\endgroup$
    – Xander Henderson
    Commented Aug 20, 2023 at 14:26

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Both are proper.

If the entire answer is $x^{-1}$ or $1/x$ then I would mark both correct and would not fuss about it. I think this sort of fuss is one of many reasons a lot of kids don't like math.

If this is part of an answer, then I think it is context dependent. Since these are fairly young kids, I'd still mark everything correct, but might put a note something like "in XXXX we usually write this as XXXX".

Of course, in a specific teaching context, you might be asking for one or the other, but you seem to be excluding those.

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  • $\begingroup$ Thanks for your thoughts. I'm not asking specifically about marking something "correct" – I was just curious if one notation was preferable (i.e., more common or more standardized) than the other. Your note about "in XXXX we usually write…" is exactly what I was after. Is there a "usual" form? Seems that there isn't. $\endgroup$ Commented Aug 20, 2023 at 13:01
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    $\begingroup$ @AlexJohnson Many teachers prefer $\frac{1}{x}$, because they want to wait until the exponential function has been introduced before putting exponents all over the place. But as long as the notation has been defined in a clear way, both notations should be fine. That is, don't use $x^{-1}$ unless you've taken at least five minutes to explain that it means $\frac{1}{x}$. Also, separate the two interrogations in your head: what you use, and what the students are allowed to use. $\endgroup$
    – Stef
    Commented Aug 20, 2023 at 13:30
  • $\begingroup$ I think there are different usual forms in different fields. $\endgroup$
    – Peter Flom
    Commented Aug 20, 2023 at 14:02
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    $\begingroup$ +1, especially for "this sort of fuss is one of many reasons a lot of kids don't like math". $\endgroup$ Commented Aug 20, 2023 at 14:03
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One of my biggest frustrations in teaching (and in speaking to other instructors, particularly elementary and high school instructors) is a belief that mathematics is prescriptivist and inscrutable. There is an idea that certain notations or processes are "right", and others are "wrong". For example,

  • Is $\frac{\sqrt{2}}{2}$ preferable to $\frac{1}{\sqrt{2}}$?
  • Is $\frac{1}{2}$ preferable to $\frac{50}{100}$?
  • Is $(x-a)(x+a)$ preferable to $x^2 - a^2$?
  • Is $\tan(\theta)$ preferable to $\frac{\sin(\theta)}{\cos(\theta)}$?

In each of these cases, I am quite certain that students would argue with me if I wrote one of the latter expressions, or asserted that any of the latter expressions were in any way superior. Indeed, nearly every semester, I have some trig student get angry at me for writing ${1}/{\sqrt{2}}$. Usually, when I drill down into why, it is, essentially, "My high school teacher told me it was wrong!"

The best thing to do (in my opinion) is to use whatever notation is most appropriate to a particular context, and to avoid optimizing early, where

  • the "most appropriate" notation is (a) whatever notation makes it easier to perform the next step of a computation, and/or (b) whatever notation most clearly communicates the key idea, and
  • "optimizing early" means rewriting an expression into a different form before knowing that the rewrite is useful or meaningful (e.g. expanding the denominator of a rational function before realizing that it needs to be added to another rational function, and having the denominator factored is going to make that easier).

So, to answer the headline question: both $x^{-1}$ and $\frac{1}{x}$ are correct. In most contexts, there is no reason to prefer one over the other, and students should be comfortable working with either form. Unless there is some context in which one form or the other is easier to work with, there is absolutely no reason that students should be told to prefer one version over another. Don't be pedantic. :/

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I would say that both are ok, but you might be careful when dealing with inverse functions, like $\sin^{(-1)}(x)$: does it mean $\frac{1}{\sin(x)}$ or does it mean $\arcsin(x)$?

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