Rational Exponents

A rational exponent is simply an exponent that's a fraction, like the $\frac23$ in $8^{2/3}$. ("Rational" here means "a ratio of whole numbers," not "sensible.") All we're really saying is that a fraction in the exponent is a root, which lets you handle roots with the exponent rules you already know. In $a^{m/n}$, the bottom number $n$ (which root to take) and the top number $m$ (what power to raise it to) each do one simple job. So $8^{2/3}$ just means "take the cube root of 8, then square it." It looks fancier than it is.

Why a fraction means a root. Last lesson, exponents counted factors, so $a^{1/2}$ sounds like nonsense: half a factor? But we can ask what it has to be if the power rule $(a^m)^n = a^{mn}$ is going to keep working. Squaring it gives $(a^{1/2})^2 = a^{1/2\cdot2} = a$. So $a^{1/2}$ is a number whose square is $a$, and that's exactly $\sqrt a$. The same argument says $a^{1/3}$ cubes to $a$, so it's $\sqrt[3]a$. Then $a^{m/n} = (a^{1/n})^m$ is just the root, raised to a power.

Evaluating. You can take the root and the power in either order, so take the root first when it makes the numbers friendlier: $8^{2/3} = (\sqrt[3]8)^2 = 2^2 = 4$. A negative sign still means reciprocal, so $16^{-3/4} = \frac{1}{(\sqrt[4]{16})^3} = \frac18$.

Simplifying. Here's where this pays off. Roots have awkward rules of their own, but once you rewrite them as exponents, the rules from the last lesson do all the work. Convert roots to exponents, simplify, and convert back if you need to.

Kepler's third law ($T \propto r^{3/2}$), metabolic scaling in biology (energy use $\propto$ mass$^{3/4}$) and musical tuning ($2^{1/12}$ per semitone) use fractional powers.