Is there a preferred way to format a negative exponent?

Question

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.

Answer 1

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.

Answer 2

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. :/

Answer 3

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 $\arcsin(x)$?