The rules of indices apply to expressions involving positive, negative, and fractional indices. These rules allow powers to be simplified and calculations to be carried out efficiently.

The rules only apply when the bases are the same.

When multiplying powers with the same base, add the indices:

$$
a^m \times a^n = a^{m+n}
$$

For example:

$$
2^3 \times 2^{-1} = 2^2
$$

When dividing powers with the same base, subtract the indices:

$$
a^m \div a^n = a^{m-n}
$$

For example:

$$
5^4 \div 5^{\frac{1}{2}} = 5^{\frac{7}{2}}
$$

When raising a power to another power, multiply the indices:

$$
\left(a^m\right)^n = a^{mn}
$$

For example:

$$
\left(4^{\frac{1}{2}}\right)^2 = 4^1
$$

A negative index represents a reciprocal:

$$
a^{-n} = \frac{1}{a^n}
$$

A fractional index represents a root:

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

These rules can be combined when simplifying expressions containing different types of indices.

You should be able to:

• Apply index laws involving positive, negative, and fractional indices
• Simplify expressions fully using index notation
• Recognise when index rules do not apply

Using the full set of index rules confidently is essential for algebraic manipulation and higher-level problem solving.