I too agree completely with Brett and Brian; "start with rational exponents and
then go to radical form."
I have some other thoughts for what to do with radicals (besides isolate them a
la Bobby Seal.)
1. Use radicals to factor x-y = (x^(1/2) - y^(1/2))(x^(1/2) + y^(1/2))
= (x^(1/n) - y^(1/n))(x^((n-1)/n)+ . . .+y^((n-1)/n))
2. Newton was inspired to develop the binomial theorem by looking at the areas
under curves of the form y = (1-xx)^n. The first terms were always x - nxxx/6
(oh dear, now I'll be filtered out of the Library of Congress) This he
generalized to the situation for n=1/2, to come up with (1-xx)^(1/2) = 1 -
(1/2)xx - (1/8)x^4 - (1/16)x^6 - . . . So he didn't use the famous triangle at
all!
3. The binomial theorem applies nicely to rational exponents. For example
(8x-y)^(1/3) = 2x^(1/3)(1-(y/8x))^(1/3)
= 2x^(1/3)(1 + (1/3)(-y/8x) + ((1/3)(-2/3)/2!)(-y/8x)^2 +
...)
4. Continued fractions!
5. Babylonian algorithm!
6. In their essay, "New Names for Old" Edward Kasner and James Newman write,
"The word radical, favorite call to arms among Republicans, Democrats,
Communists, Socialists, Nazis, Fascists, Trotskyites, etc., has a less hortatory
and bellicose character in mathematics. <Not according to some of my
students:~b> For one thing, everybody knows its meaning: i.e., square root, cube
root, fourth root, fifth root, etc. Combining a word previously defined with this
one, we might say that the extraction of a root is the evolution of a radical.
The square root of 9 is 3; the square root of 10 is greater than 3, and the most
famous and the simplest of all square roots, the first incommensurable number
discovered by the Greeks, the square root of 2 is 1.414. . . . . There are also
composite radicals - expressions like 7^(1/2) + 10^(1/5). The symbol for a
radical is not the hammer and sickle, but a sign three or four centuries old, and
the idea of the mathematical radical is even older than that. The concept of the
"hyperradical," or "ultraradical," which means something higher than a radical,
but lower than a transcendental is of recent origin.
They go on to talk about these weird numbers which are apparently used to
represent roots of quintics which can't be written as regular radicals... Very
interesting, but my question is re the reference to radical notation as being 3
or 4 centuries old. I've read that Oresme (1323 - 1382) introduced the notion,
but not the notation, of a rational exponent. Nicolas Chuquet (ca. 1500) used
notation like R)².3 for the square root of 3, but as far as I know, not even
Cardan nor Tartaglia used modern radical notation. From whence to it sprout?
Geoff Hagopian
Palm Desert, CA
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