Indices

Indices are the little numbers written above a number. They tell you how many times to multiply a number by itself.

For example:

$$2^3=2\times 2\times 2=8$$

• $2$ is the base.
• $3$ is the index (or power).
• $8$ is the answer.

The most important thing for GCSE Maths is learning the rules of indices so you can simplify expressions quickly.

The rules you need to know

Multiplying powers with the same base

Add the indices.

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

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

Dividing powers with the same base

Subtract the indices.

$$a^m÷a^n=a^{m-n}$$Example:

$$5^6÷5^2=5^4$$

A power raised to another power

Multiply the indices.

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

$$(3^2{)}^4=3^8$$Because $2\times 4=8$

Power of zero

Any non-zero number to the power of zero equals 1.

$$a^0=1$$Example: $7^0=1$

Negative indices

A negative index means take the reciprocal.

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

$$2^{-3}=\frac{1}{2^3}=\frac{1}{8}$$

Fractional indices

The bottom number is the root; the top number is the power.

$$a^{1\mathrm{/}2}=\sqrt{a}$$Example:

$${16}^{1\mathrm{/}2}=\sqrt{16}=4$$Another example:

$$8^{1\mathrm{/}3}=\sqrt[3]{8}=2$$

Fractional indices with a power

Find the root and then raise it to the power.

$$a^{m\mathrm{/}n}=(\sqrt[n]{a}{)}^m$$Example:

$${27}^{2\mathrm{/}3}=(\sqrt[3]{27}{)}^2=3^2=9$$

How to answer GCSE indices questions

Let’s go through some typical exam questions.

Example A: Simplify $x^4\times x^7$

Step 1: Check the bases are the same. Both are $x$x.

Step 2: Add the indices.

$$x^4\times x^7=x^{4+7}$$Answer: $x^{11}$

Example B: Simplify $a^9÷a^3$

Step 1: The bases are the same.
Step 2: Subtract the indices.

$$a^9÷a^3=a^{9-3}$$Answer: $a^6$

Example C: Simplify $(y^3{)}^5$

Step 1: Recognise a power raised to another power.
Step 2: Multiply the indices.

$$(y^3{)}^5=y^{3\times 5}$$Answer: $y^{15}$

Example D: Evaluate ${81}^{3\mathrm{/}4}$

Step 1: The denominator is 4, so find the fourth root.

$$\sqrt[4]{81}=3$$Step 2: The numerator is 3, so cube the answer.

$$3^3=27$$Answer: $27$

Common GCSE mistakes to avoid

Important: You can only add or subtract indices directly when the bases are the same.

For example, $2^3\times 3^2$ cannot be simplified by adding 3 and 2 because the bases are different.

Quick revision sheet

Before your GCSE exam, memorise these five rules first:

• Multiply powers with the same base → add indices.
• Divide powers with the same base → subtract indices.
• Power of a power → multiply indices.
• Zero index → answer is 1, for any non-zero base.
• Negative index → take the reciprocal.

Then learn fractional indices, where the denominator tells you the root and the numerator tells you the power.
If you can apply these rules confidently, you’ll be able to answer a large proportion of GCSE indices questions.