I guess what I was trying to say is that treating that as a ternary relation is already overcomplicating the matter. Really, there's no need for nth root notation at all, because we already have fractional exponentiation. Why have a third place on the triangle putting any n there is the same as putting 1/n on the top?
Part of the argument for the triangle is that the relationships you'd have to learn are suggested by simple geometric manipulations.
But why learn whole new geometric manipulations when you've already learned the algebraic manipulations behind multiplication and division? By converting exponentiation and logarithms into infix notation, you do away with one side of the triangle and you can learn the relationships by just thinking "it's exactly like multiplication and division, except [etc]"
Incidentally, sure, we could just pick ^ to be our basic operator and define everything else in terms of that. Well, just as well, we could define everything in terms of logarithm. Or everything in terms of roots! In fact, one might ask why you are so happy to define a logarithm operator v to undo ^ in one direction, but do not want ever to define and work with a root operator to undo ^ in the other direction.
Is it because we can get away with saying a^(1/n) instead of n-th root of a? Well, just as well, why should we have a division operator? Instead of a/b, we can say a * b^(-1).
Or perhaps we oughtn't have multiplication, or exponentiation with arbitrary bases. Perhaps we should just have addition, negation, natural logarithm, and the natural exponential. Then we define a * b as exp(ln(a) + ln(b)), define b^c as exp(c * ln(b)), etc.
But, of course, just because F can be expressed in terms of G doesn't mean it's always convenient or pertinent to think of F that way; sometimes it's pertinent to think of F just qua F, as an atomic entity in its own right. And, thus... the proliferation of different functions people talk about.
Though this does mean people then have to keep track of what all these different functions are defined as, and trace back through how that makes them relate. So... I don't know. I don't have all the answers. Yet.
Why think of ternary relations? Well, if you do not recognize that there is an "Any two variables here determine the third" structure here, you are missing a fruitful insight into the matter. That's why. Sometimes that's a useful perspective, sometimes perhaps less so.
That can be useful not only for exponentiation/logarithm/roots, incidentally. It can be useful to recognize with addition/subtraction/subtraction, or multiplication/division/division, as well. These all have a similar structure: three values in a relationship, such that any two determine the third. The triangle is supposed to help recognize this structure, though it helps with nothing else on top of that.
"By converting exponentiation and logarithms into infix notation, you do away with one side of the triangle and you can learn the relationships by just thinking 'it's exactly like multiplication and division, except [etc]'"
The problem is, there's rather a LOT of "except". Standard * is commutative, while ^ isn't. * is associative, while ^ isn't. * turns additions on either side into additions of the result, while ^ turns multiplications on left into multiplications of the result, but multiplications on the right only into chained operations (i.e., a ^ (b * c) = (a ^ b) ^ c), while additions on the right are turned into multiplications of the result (i.e., a ^ (b + c) = (a ^ b) * (a ^ c)), and additions on the left are turned into binomial theorem expansions!
These are big differences! There are particular similarities too, of course. There's a mix of similarities and lots of differences. What that amounts to pedagogically is... a muddle.
Anyway, the triangle manipulations in the video aren't at all compelling to me, but regardless, the ternary relation structure IS useful to recognize (that any two out of three determine the third, and this is all the various traditional functions do), and again, this is something to be recognized not just for powers, logarithm, etc., but even for our familiar addition/subtraction or multiplication/division problems.
Part of the argument for the triangle is that the relationships you'd have to learn are suggested by simple geometric manipulations.
But why learn whole new geometric manipulations when you've already learned the algebraic manipulations behind multiplication and division? By converting exponentiation and logarithms into infix notation, you do away with one side of the triangle and you can learn the relationships by just thinking "it's exactly like multiplication and division, except [etc]"