The other day, I introduced my students to fractional or rational exponents. First thing my contrary student asked "When is this really used?" Off the top of my head, I stated you need it when you have different roots and you have to combine them. It also makes it easier when differentiating in Calculus but other than that? I don't know.
I worked before I became a teacher but the jobs I had did not require the use of exponents in any way so I had trouble answering his question.
It is used more than I realized and I've taught some of these uses without understanding I was using fractional exponents.
1. Financial industry. They use these types of exponents when they calculate compound interest. Fractional exponents are also used to calculate depreciation or calculating the increased value of a home.
Biologists use rational exponents to calculate surface areas of different animals as a way of comparing sizes using the formula S = km^2/3 where k is a constant and m represents the mass of the animal.
In music, the exact frequency is found using a rational exponent. The formula is f = 440 *2^1n/2 where n is the number of black and white keys above or below the 440 frequency.
Furthermore, the power generated for a ship uses rational exponents in the formula
P = (d^2/3 *s^3)/c. d represents the ships displacement, s is the speed in knots and c is the Admiralty coefficient.
I saw it used in a problem comparing the surface area of spheres which I'd never thought of.
For all the apparent uses in real life, its very hard to find concrete examples for uses other than the few I was able to actually find. I know for me, I see the use as being able to combine radicals of different roots with the same bases which indicates you are combining lengths to get a total. Maybe in carpentry or trig?
I would love to have others contribute examples so I can share them all with my students. It is frustrating when you look and just cannot find that many
Thank you in advance.