Fractional Indices

Where fractional indices come from

The index laws you already know have to keep working. Since $a^{\frac{1}{2}} \times a^{\frac{1}{2}} = a^{\frac{1}{2} + \frac{1}{2}} = a^1$, the number $a^{\frac{1}{2}}$ multiplied by itself gives $a$ — so it must be $\sqrt{a}$. In the same way $a^{\frac{1}{3}} \times a^{\frac{1}{3}} \times a^{\frac{1}{3}} = a$, so $a^{\frac{1}{3}} = \sqrt[3]{a}$.

Any fraction as an index

Split the fraction using the power law $(a^m)^n = a^{mn}$:

$a^{\frac{m}{n}} = \left(a^{\frac{1}{n}}\right)^m = \left(\sqrt[n]{a}\right)^m$

The denominator is the root; the numerator is the power. Do the root first: $8^{\frac{2}{3}} = 2^2 = 4$ is far easier than $\sqrt[3]{64}$.

Negative fractional indices

A negative index still means “one over”, so there are three jobs: flip, root, power. With a fraction as the base, flipping turns it upside down.

Solving equations with indices

Two methods come up again and again:

• Same base: write both sides as a power of the same number and compare the indices. $4^x = 32 \Rightarrow 2^{2x} = 2^5 \Rightarrow x = 2.5$.
• Reciprocal power: to undo $x^{\frac{m}{n}}$, raise both sides to $\frac{n}{m}$.