The rules of indices allow calculations with powers to be carried out efficiently. These rules apply when working with numbers written in index form using positive integral indices.

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

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

For example:

$$
2^3 \times 2^4 = 2^7
$$

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

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

For example:

$$
5^6 \div 5^2 = 5^4
$$

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

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

For example:

$$
\left(3^2\right)^3 = 3^6
$$

These rules only apply when the bases are the same.

You should be able to:

• Apply index rules to simplify expressions
• Perform calculations involving powers accurately
• Recognise when index rules can and cannot be used

Using the rules of indices correctly is essential for efficient calculation and algebraic manipulation.