Indices (AQA GCSE Maths): Revision Notes

Indices

Indices are a mathematical way to represent repeated multiplication. They include square roots, cube roots and powers. Understanding the index laws is essential for simplifying mathematical expressions and solving complex problems.

What are indices?

Indices (also called powers or exponents) show how many times a number is multiplied by itself. In the expression $a^n$, 'a' is called the base and 'n' is called the index or power.

For example: $6 \times 6 \times 6 \times 6 \times 6 = 6^5$

The seven index laws

Law 1: Multiplying powers with the same base

When you multiply powers that have the same base, you add the indices.

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

Example: $4^3 \times 4^7 = 4^{3+7} = 4^{10}$

This works because you're combining all the repeated multiplications together.

Law 2: Dividing powers with the same base

When you divide powers that have the same base, you subtract the indices.

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

Example: $12^6 \div 12^3 = 12^{6-3} = 12^3$

This works because division cancels out some of the repeated multiplications.

Law 3: Power of a power

When you raise a power to another power, you multiply the indices.

Formula: $(a^m)^n = a^{mn}$

Example: $(7^3)^5 = 7^{3 \times 5} = 7^{15}$

This happens because you're repeating the original power multiple times.

Law 4: Negative powers

A negative power creates a reciprocal (fraction with 1 on top).

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

Example: $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

Law 5: Reciprocals

The power of -1 gives you the reciprocal of the number.

Formula: $a^{-1} = \frac{1}{a}$

This means that $a^{-1}$ is the reciprocal of a. You can find the reciprocal of a fraction by turning it upside down.

Example: $\left(\frac{5}{9}\right)^{-1} = \frac{9}{5}$

Law 6: Powers of fractions

When you raise a fraction to a power, you apply the power to both the numerator and denominator.

Formula: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Example: $\left(\frac{3}{10}\right)^2 = \frac{3^2}{10^2} = \frac{9}{100}$

Law 7: Combining rules

You can apply multiple rules in the same calculation, working through them one step at a time.

Example: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n = \frac{b^n}{a^n}$

More complex example: $\left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3 = \frac{3^3}{2^3} = \frac{27}{8}$

Powers of 0 and 1

These have special properties that you need to remember:

Power of 0

Power of 1

Worked example

Exam tips