Betsy 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,

256^(-3/4) = [(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.