Fractional indices are another way of writing roots and powers. They allow expressions involving roots to be written using index notation.

A fractional index represents a root and a power.

An index of \( \frac{1}{2} \) represents a square root:

$$
a^{\frac{1}{2}} = \sqrt{a}
$$

An index of \( \frac{1}{3} \) represents a cube root:

$$
a^{\frac{1}{3}} = \sqrt[3]{a}
$$

More generally, for a fractional index:

$$
a^{\frac{1}{n}} = \sqrt[n]{a}
$$

When the numerator is greater than 1, it represents a power as well as a root:

$$
a^{\frac{m}{n}} = \sqrt[n]{a^m}
$$

For example:

$$
16^{\frac{1}{2}} = 4
$$

$$
27^{\frac{2}{3}} = \sqrt[3]{27^2} = 9
$$

Fractional indices provide a compact way to write expressions involving roots and are commonly used when simplifying algebraic expressions.

You should be able to:

• Interpret fractional indices as roots
• Convert between root form and index form
• Use fractional indices correctly in calculations

Understanding fractional indices links index laws directly to surds and roots.