Rational Exponents

a^m/n means root-then-power: (√[n]a)^m.

The idea

A rational exponent extends powers from whole numbers to fractions. Repeated multiplication gives no meaning to $a^{1/2}$ — there is no half copy of a number to multiply — so we define the symbol by requiring the exponent rules to keep working. Raising a power to a power multiplies the exponents, so $a^{1/n}$ must satisfy

$\left(a^{1/n}\right)^{n} = a^{n \cdot \frac{1}{n}} = a.$

Therefore $a^{1/n}$ is the number whose $n$-th power is $a$, which is the $n$-th root $\sqrt[n]{a}$. Adding a whole-number power gives the general definition, for $a > 0$:

$a^{m/n} = \left(\sqrt[n]{a}\right)^{m}.$

The denominator names the root to take and the numerator the power to raise it to. Either order gives the same value, but taking the root first keeps the numbers small. A negative exponent still means the reciprocal: $a^{-m/n} = \frac{1}{a^{m/n}}$.

Ways to work on it

• Walkthrough. Why a^1/n must be the n-th root, and how a^m/n follows.
• Practice. Evaluate fractional (and negative-fractional) powers.
• Hardest. Solve an equation of the form x^m/n = k.

Not sure where to start? Take the ten-question placement test.