I'm helping my younger sister for her math class. She has recently been taught integer exponents, and has starteed studying radicals (mainly square roots). The next topic will be rational exponents, which she has already read a bit up on.

It's been a long time since I've learnt all this and it has become second nature to me. In particular I've never been fond of radical notation and often end up writing $(expr)^\frac{1}{2}$ instead of $\sqrt{expr}$. This means that while I can give her a good amount of help, I am not always able to justify why things work how they do and what's the motivation behind what she's taught.

Right now she understands integer exponents well; adding, multiplying them, even when rational bases are involved. Roots however confuse her, which means that while she is able to solve e.g. $y^{5/2} (\frac{x}{2})^2 \frac{x}{x^{1/2} y^{7/4}} = \frac{1}{4} x^{5/2} y^{3/4}$, the same thing written with roots $\sqrt{y^5} (\frac{x}{2})^2 \frac{x}{\sqrt{x} \sqrt[4]{y^7}}$ is unclear to her. She now has to relearn every formula she knows like $x^n y^n = (xy)^n$ written with roots $\sqrt{x}\sqrt{y} = \sqrt{xy}$.

I've explained to her that roots and rational exponents are roughly the same and how to rewrite roots as rational exponents ($\sqrt[q]{x^p} = x^\frac{p}{q}$). She isn't allowed to do that in class however, as learning about roots is mandatory, and I'm not actually trying to help her skip the subject.

But since the exponent notation is so much simpler to her (and to me) she asked me what was the reason to use roots in the first place, and I wasn't able to answer. So here's the question: what are the reasons to use roots ? Are there significant cases where the last equation I wrote above does not hold ? Are nth-roots and rational exponents actually different beasts ?

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How do you define the meaning of $x^{1/n}$ in the first place, if not as the $n$th root of $x$? –  Henning Makholm Oct 16 '12 at 20:06
@Henning: It’s the same definition whether you call it $x^{1/n}$ or $\sqrt[n]{x}$. It seems to me that the question is really as much about notation as about the mathematics involved. –  Brian M. Scott Oct 16 '12 at 20:09
@Henning: This is a response to something that I never said. Of course you need a distinct definition of $a^b$ when $b$ is not an integer; so what? As I said, for the case $b=\frac1n$ it’s the same definition whether you call it the $n$-th root or the $\frac1n$-th power. –  Brian M. Scott Oct 16 '12 at 21:36
... The only reason I can see for thinking that $x^{1/n}$ is any easier to understand than $\sqrt[n]{x}$ is that one falsely believes that everything one knows about power laws and so forth for the integer-exponent case can be transferred formally to the fractional-exponent case and this is not true. It is a new thing that needs new definition, new proofs, and new insight, and any possible "oh, I already knows this" feeling that comes from the apparently-familiar $x^{1/n}$ notation is a false sense of safety that just deceives them into thinking there's nothing new to learn. –  Henning Makholm Oct 17 '12 at 10:59
$1 = 1^{\frac{1}{2}} = ((-1)^2)^{\frac{1}{2}} = (-1)^{2\times \frac{1}{2}} = (-1)^1 = -1$ And this is why they are introduced with the radical. –  xavierm02 Oct 17 '12 at 12:12

To attempt to answer the actual question, I think that it’s a largely two-fold consequence of the historical inertia mentioned in the comments by @coffeemath. On the one hand, it’s simple classroom inertia: it’s ‘always’ been done this way, so we do it this way. On the other hand it’s the practical consideration that since the radical notation does survive in real-world use, students need to learn how to deal with it. None of this, however, justifies the requirement that students deal with it directly, rather than by translating it into a less cumbersome, more easily manipulated notation. Indeed, in my view this is a good occasion to make the point that well-chosen notation makes our mathematical lives easier.

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In my opinion, radical notation is still around for one reason, and one reason only - namely, because we don't have good notation for reciprocation. For instance, as you point out in your answer, its well-known that

$$x^{1/n} = \sqrt[n]{x}$$

The problem is, the left hand side looks complicated - after all, it involves a fraction, and we've got this redundant $1$ floating around - so this is basically pedagogical suicide. So, quite reasonably, educators decided to stick with the notation on the left-hand side. After all, fractions are hard.

But the problem is solved if we just had better notation for reciprocation. For example, lets imagine for a moment that we live in a world where fractions are written upside down. So for example, $$\frac{4}{3}$$

means 'into $4$, divide $3$.' More conceptually, we can image that it takes 4 steps to walk the length of a unit, and we're taking 3 of those steps. So we haven't quite walked one whole unit. Anyway, in this notational paradise, we can simply define that the reciprocal of a number $n$ is $\underline{n}.$ So that way $$\frac{4}{3} = \underline{4}\cdot 3.$$

Furthermore, in this hypothetical notational paradise, we have not only that $x^2$ is the square of $x$, but also that $x^\underline{2}$ is the square-root. Indeed, we may write: $$x^\underline{n} = \sqrt[n]{x}$$

which avoids the aforementioned pedagogical trap. So, I conjecture that, if we lived in this hypothetical paradise, educators would have boycotted the $\sqrt[n]{x}$ notation a long time ago. Unfortunately, we do not, and that is why radical notation persists.

On a somewhat related topic, I think that fractions are taught completely wrong. The problem is, the division operation $/$ has terrible properties. For example, its not associative, and expressions likes $2/(3/4)$ are initially confusing to students. Furthermore, its not commutative, so we end up with a whole slew of confusing facts to remember, like $x/1$ always equals $x$ but $1/x$ generally does not equal $x$. So, I would argue that, as opposed to fraction notation, we should be teaching products and reciprocals.

For the remainder of the discussion, lets return to notational paradise world. So if we're wanting to teach rational numbers, we can proceed as follows.

1. Explain products of natural numbers using two metaphors. Firstly, a geometric one, whereby $a \cdot b$ is the number of squares in a grid that is $a$ squares high and $b$ squares wide. Secondly, a 'walking' metaphor, whereby $a \cdot b$ is the number of units walked if our steps are of length $a$ and we take $b$ of them. Thankfully, both these metaphors generalize to continuous numbers.

2. Explain reciprocals using both metaphors. The geometric metaphor: We start off with a 1x1 box. If we stretch it so that it now has width $10$, what length does it need to have in order so that its area remains unchanged? That length is called the reciprocal of $10$. The walking metaphor: Suppose it takes $10$ steps to walk one whole unit. How long is each step? That length is called the reciprocal of $10.$

3. Use both metaphors to explain why $\underline{n} \cdot n = 1.$ Ask the kids what the reciprocal of $1$ is and have them puzzle it out themselves.

4. Explain the idea of simplifying expressions involving products and reciprocals. Like $$\underline{3} \cdot 9 = \underline{3} \cdot 3 \cdot 3 = (\underline{3} \cdot 3) \cdot 3 = 1 \cdot 3 = 3$$

5. Go back to the metaphors again to explain things like $\underline{ab} = \underline{a}\cdot\underline{b},$ as well as why the reciprocal of the reciprocal of $x$ is still $x$. Give the students more difficult simplification problems that require the use of these new identities.

6. After a year or two, introduce the notation $\frac{a}{b}$ as shorthand for $\underline{a} \cdot b,$ but only as preparation for learning how to add fractions. Explain concepts like $$\frac{ab}b = \underline{a}$$ by appealing to the properties of products and reciprocation that students are already very familiar with.

7. Introduce fractions involving negative numbers only at the very end, once a high level of proficiency with manipulating positive numbers has already been achieved.

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The problem is that at this level, things are not usually thaught because of their mathematical beauty but rather because of their practical use. And division is everywhere, and the world doesn't really care that it's not commutative. If you say to me that when you have $20$ pieces of chocolate to distribute among $4$ people you multiply $20$ by the reciprocal of $4$, I don't believe you. –  Javier Badia Aug 2 '13 at 2:13
@JavierBadia, can you clarify as to which part of my answer your comment is directed? –  user18921 Aug 2 '13 at 4:31
The second part, mostly. The whole thing about teaching fractions differently. –  Javier Badia Aug 2 '13 at 15:00
@JavierBadia, mathematical beauty goes hand-in-hand with teachability. So even if your only concern is preparing kids to be cogs in the economic machine - an acceptable goal, in my opinion - you have to at least admit that the current way of teaching fractions isn't doing a very good job of it. So look, if there's some specific points I make that you disagree with, then I'd be interested to hear your opinions and ideas. But if you're trying to argue that the current way of teaching fractions is acceptable, we're going to have to agree to disagree. –  user18921 Aug 4 '13 at 12:48
I would argue that something like what's outlined in Unpacking A Conceptual Lesson: The Case of Dividing Fractions (sci.sdsu.edu/CRMSE/IMAP/pubs/…) is really close to the way you'd like fractions to be taught (still different than many books/teachers), while still using standard notation. I disagree with the claim that "the [common] current ways of teaching fractions is flawed" implies "we should switch notations when first teaching fractions". I think @JavierBadia's example supports "we shouldn't switch", and I agree with the antecedent... –  Mark S. Nov 9 '13 at 2:29
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