I seem to recollect that Dr. Peterson has available a scolarly source on the history of mathematical notation. Just based on wikipedia, the radical symbol in its modern guise was developed by Descartes and seems to have originally been just a square root symbol. Furthermore, even today, the symbol means square root unless explicitly specified as being a higher order root. So, unlike Halls, I have no problem with LaTeX's use of \sqrt to indicate the symbol of radical plus attached vinculum. LaTeX is typographical software rather than mathematics, and I do not find it difficult to keep straight the difference between the name of a symbol and the name of the concept symbolized.
Like Halls, however, I find it it very sloppy to read "third root of x" as "third square root of x," which is literally meaningless and may be misconstrued as the cube root of the square root. Subhotosh' and cubist's jokes nicely reinforce Hall's fundamental point. Of course, in the case of the OP, this was almost certainly not sloppiness but simple lack of familiarity with the notations. So it was a good question.