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There is a good history of mathematical symbols here: http://jeff560.tripod.com/operation.html
Here is what that site says about the square root symbol:
Square root. The first use of a capital R with a diagonal line was in 1220 by Leonardo of Pisa in Practica geometriae, where the symbol meant "square root" (Cajori vol. 1, page 90).
The radical symbol first appeared in 1525 in Die Coss by Christoff Rudolff (1499-1545). He used the symbol (without the vinculum) for square roots. He did not use indices to indicate higher roots, but instead modified the appearance of the radical symbol for higher roots.
It is often suggested that the origin of the modern radical symbol is that it is an altered letter r, the first letter in the word radix. This is the opinion of Leonhard Euler in his Institutiones calculi differentialis (1775). However, Florian Cajori, author of A History of Mathematical Notations, argues against this theory.
In 1637 Rene Descartes used , adding the vinculum to the radical symbol La Geometrie (Cajori vol. 1, page 375).
Placement of the index within the opening of the radical sign was suggested in 1629 by Albert Girard (1595-1632) in Invention nouvelle. He suggested this notation for the cube root (DSB; Cajori vol. 1, page 371).
According to Cajori (vol. 1, page 372) the first person to adopt Girard's suggestion and place the index within the opening of the radical sign was Michel Rolle (1652-1719) in 1690 in Traité d'Algèbre.
However, a history note in a high school textbook states that the symbol was first used by Girard “around 1633” (UCSMP Advanced Algebra, 2nd ed., 1996, page 496).
According to Tropfke, Geschichte der Elementarmathematik, 4th edition, vol. 1: Arithmetik und Algebra, Berlin/New York: De Gruyter, 1980, p. 296, Girard introduced this use with examples for the cubic and the quintic root on fol. 1r° of the Invention nouvelle and suggested on the same page also (3/2)49 for the fractional exponent of 49 [3/2 written as fraction]. [This information contributed by Siegmund Probst.]
Examples of Leibniz's use of the radical sign with index placed in the opening:
VII,3 N. 57_2 p. 736 (End of April 1676)
VII,3 N. 58 p. 752 (May 3, 1676)
VII,3 N. 60 p. 760 (June 26, 1676)
In the Mathematical Gazette of Feb. 1895, G. Heppel wrote, "Following Chrystal, Todhunter, Hall and Knight, and the majority of writers [sqrt]a should be considered a quantity having one and not two values, although the algebra of C. Smith and the article by Professor Kelland in the Encyclopedia Britannica make [sqrt]a have two values."