Nested radicals, smoothness, and simplification

I saw an expression involving a nested radical, namely

.

(Write φ = (1 + φ)^1/2 and solve for φ.) The Wikipedia article on nested radicals led me to Simplifying Square Roots of Square Roots by Denesting. The authors tell us that:

The term surd is used by TeX as the name for the symbol √ Maple has a function called surd that is similar to the nth root defined here; like all good mathematical terms, the precise definition depends upon the context. In general, a mathematical term that does not have several conflic! ting definitions is not important enough to be worth learning.

This reminds me of a couple things that happened in my class yesterday. First, I was defining what it means for a curve to be smooth; our definition was that the curve given by the vector function r(t) is smooth if r'(t) is continuous and never zero, except perhaps at the endpoints of the interval over which it's defined. (This makes smoothness a property of a parametrization, which is a bit counterintuitive. I suppose that one could define a curve -- as an abstract set of points -- to be smooth if it has a smooth parametrization. Although I haven't worked it out, I assume that if a curve has a smooth parametrization, the arc-length parametrization is smooth.) One of the students said "but the professor said 'smooth' means something else!" I'm not sure if the professor actually said "smooth means X" or if he said "some people think smooth means X", but i! t's a good point. (In particular, "smooth" often seems to mea! n that a function has infinitely many continuous derivatives.)

Second, the article is about using computer algebra systems to simplify expressions like

where the left-hand side is "simpler"; sometimes my students worry that they are not presenting their answer in the simplest form. While I'll accept any reasonably simple answer (unless the problem statement specifies a particular form), it is remarkably difficult to define what "simple" means.

One rule I have figured out, though, is that 4x - 4z - 8 = 0 should be simplified to x - z - 2 = 0 by dividing out the common factor. In general, given a polynomial with rational coefficients, one probably wants to multiply to clear out the denominators and then divide by an! y common integer factor of the new coefficients, so the resulting coefficients are relatively prime integers. The article addresses this sort of "canonicalization" in the context of nested radicals. I keep telling my students that they should keep that sort of thing in mind, especially since our tests will be mostly multiple-choice.

(Sometimes I'm tempted to define "simplest" as "requires the fewest symbols"... but how does one prove that some 100-character expression one has written can't be written in 99 characters? And how do you count something like "f(x, y)= (x+y)^1/2 - (x-y)^1/2, where x = foo and y = bar?" ("foo" and "bar" are supposed to be very complicated expressions.) Do you plug foo and bar into the original equation and then count the characters, or do you count the actual characters that are between the quotation marks?)

polynomial simplifier