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What the heck are fraction exponents?

Fractional exponents (a.k.a. rational exponents) are a little weird. They force us to think backwards, to ask, "What number multiplied by itself yields the base?" If this questions sounds familiar, it's because we ask the same question when figuring out square roots (and other roots). Rational exponents are just another, calculator-friendly way of expressing roots.

If we want to find 8 to the 1/3rd power, we ask, "What number multiplied by itself 3 times equals 8?" And then take one of them.

If we want to find 8 to the 2/3rd power, we ask, "What number multiplied by itself 3 times equals 8?" And then we take 2 of them (and multiply together).

Fractional exponents are really useful, especially when figuring out the rate in exponential growth equations. It feels easier remembering how to raise a number to, say, ^(1/5) than it does finding that x root button on a calculator. I never know where it's hiding or if I am even using it right when I do find it! So knowing that a square root is the same as ^(1/2), etc. has made my math life easier.

Just yesterday I saw an ad online declaring that if a 25 year old saved \$5/day, at age 65 they'd have almost \$300,000 saved in those 40 years. What would the approximate interest rate be if this were true?

Let's see... \$5/day = \$1825/year for 40 years...

300000 = 1825(1 + r)^40

164.38 = (1 + r)^40

164.38^(1/40) = 1 + r

1.136 = 1 + r

0.136 = r

13.6% interest rate. Hmmmm...

My calculations are a bit off because I assumed interest compounding annually and banks usually compound interest more regularly, but even so, a guaranteed annual interest rate of almost 14% for 40 years? There's just no way anyone can guarantee a return like that (unless you're Bernie Madoff maybe). How many people saw that ad and believed it? How many knew how to use rational exponents to solve for r?