[This is the second of several math-related pieces I plan to post on this
web site. One of the big stumbling blocks for high school algebra students -- and, it appears, some professional logicians!
-- is the issue of fractional exponents. What do those little numbers mean that we put above and to the right of numbers and
letters? I hope this short piece helps to strip away some of the mystery surrounding these little beasties. Roger E. Bissell,
September 4, 2002]
Lichtenberg, an interested reader of my earlier paper on negative exponents, wrote the following:
"How about fractional
exponents? How can a number with a fractional exponent be expressed (in English) in reference to the unit 1 as you do for
positive, negative, and zero exponents?
"To express the unit 1's relationship to positive and negative
exponents, you say, 'For any real number, r, a positive exponent, n, indicates that the unit 1 is to be considered as
having been multiplied by r a total of n times. A negative exponent, -n, indicates that the unit 1 is considered as having
been divided by r a total of n times.'
"To express the unit 1's relationship to zero exponents, you
say, 'The unit 1 should be considered ... as not having been multiplied (or divided) by the base number any times.'
"OK, so take x raised to the 1/n power. It is equal to the nth root of x. A fractional exponent -- specifically,
an exponent of the form 1/n -- means to take the nth root instead of multiplying or dividing. For example, 4(1/3) is the 3rd
root (cube root) of 4.
"Obviously, the unit 1 is explicit here in the numerator of the fraction, but how would
you rewrite the definition of a fractional exponent "an exponent of the form 1/n -- means to take the nth root"
and express this in terms of how the unit 1 is central to the operation being performed in a fashion similar to how you expressed
the unit 1's centrality in the cases of positive, negative, and zero exponents?"
OK, I scratched my head a bit,
and I figured it out!
For any real number, r, a positive fractional exponent, 1/n, indicates that the unit 1 is
to be considered as having been multiplied by one of r's n equal factors. A negative
fractional exponent, -1/n, indicates
that the unit 1 is considered as having been divided by one of r's n equal factors.
For example if r = 8 and
n = 3, we want to express 8 to the 1/3 power (i.e., the cube root of 8) like this: 1 multiplied by one of 3 equal factors
of 8. And if r = 256 and n = 4, we want to express 256 to the -1/4 power (i.e., the negative fourth root of 256) like this:
1 divided by one of 4 equal factors of 256.
The arithmetic is: 8^(1/3) = 1 x 2 = 2,
256^(-1/4) = 1/4
All right, let's push on. What about mixed fraction exponents? For instance, 8 to the 2/3 power, or 256 to the -3/4
power. Well, the former is 1 multiplied by one of 3 equal
factors of 8 a total of 2 times, and the latter is 1 divided
by one of 4 equal factors of 256 a total of 3 times.
The arithmetic is: 8^(2/3) = 1 x 2 x 2 = 4,
= [(1/4)/4]/4 = 1/(4 x 4 x 4) = 1/64
The principles for fractional exponents are thus seen to be analogous to those
for integral exponents, and those for mixed fractional exponents are a combination of the
other two. So, to generalize
for any rational exponent: For any real number, r, a positive rational exponent, m/n, indicates that the unit 1 is to be considered
as having been multiplied by one of r's n equal factors a total of m times. A negative rational exponent, -m/n, indicates
that the unit 1 is considered as having been divided by one of r's n equal factors a total of m times.
As in my previous
paper on exponents, I apologize to the mathematicians in the group for the inelegance of this explanation. I hope it is clear
enough for everyone else.