Common Mistakes With Indices in GCSE Maths

Indices questions often look short. That is partly why they can be dangerous. A student sees two powers, remembers something about adding or multiplying, and writes an answer before checking which rule applies.

The most common mistakes with indices in GCSE Maths come from using a correct rule in the wrong situation. The solution is not to memorise more disconnected facts. It is to recognise the structure of the expression first: are you multiplying, dividing, raising a power to another power, or dealing with a negative or fractional index?

Imagine a student correcting the errors that have been costing them marks. Instead of simply crossing out each answer, they identify why the mistake happened and attach the right rule to it. Here is the method they can follow.

A quick indices checklist

Before simplifying an expression, ask:

• Are the bases the same?
• Is the operation multiplication or division?
• Is one power being raised to another power?
• Are there coefficients outside the powers?
• Is the index zero, negative or fractional?
• Do brackets change what the index acts on?

The main laws are:

$$a^m\times a^n=a^{m+n}$$ $$\frac{a^m}{a^n}=a^{m-n}$$ $$(a^m{)}^n=a^{mn}$$ $$a^0=1$$ $$a^{-n}=\frac{1}{a^n}$$

For division, zero indices and negative indices, the base must be non-zero where required. You can review these rules in the Maths Genie Indices lesson and the accompanying Indices revision guide.

A student stops revision autopilot from taking the pencil

Common indices mistakes and how to fix them

Adding or multiplying the indices at the wrong time

Suppose the student sees:

$$x^3\times x^5$$

They write $x^{15}$ because they multiply $3$ and $5$. The confusion is understandable because multiplying indices is sometimes correct, but only when a power is raised to another power.

Here, powers with the same base are being multiplied, so add the indices:

$$x^3\times x^5=x^{3+5}=x^8$$

The expanded form explains why:

$$x^3\times x^5=(x\times x\times x)(x\times x\times x\times x\times x)=x^8$$

Use the question's operation as your cue. Multiplication between powers means add the indices, provided the bases are the same.

Adding indices when raising a power to a power

Next, the student meets:

$$(y^4{)}^3$$

They write $y^7$, treating it like $y^4\times y^3$. But the outer index means that the entire expression $y^4$ appears three times:

$$(y^4{)}^3=y^4\times y^4\times y^4=y^{4+4+4}=y^{12}$$

Therefore:

$$(y^4{)}^3=y^{4\times 3}=y^{12}$$

The correction is simple: when a power is raised to another power, multiply the indices.

Applying an index to only part of a bracket

Consider:

$$(3x{)}^2$$

A frequent answer is $3x^2$. This forgets that the index applies to everything inside the brackets:

$$(3x{)}^2=(3x)(3x)=9x^2$$

Compare that with:

$$3x^2$$

Here, only $x$ is squared. Brackets are not decoration. They tell you exactly what the index controls.

You now have the recognition method in hand, so use the Maths Genie Lessons and Question Bank for Indices to practise choosing the rule before carrying out the arithmetic.

Trying to use an index law across addition

The student then sees:

$$x^2+x^3$$

They are tempted to write $x^5$. However, the rule for adding indices applies when powers with the same base are multiplied, not when the terms are added.

The terms $x^2$ and $x^3$ are not like terms, so they cannot be collected. The expression remains:

$$x^2+x^3$$

It could be factorised as:

$$x^2(1+x)$$

That is a factorising step, not an index law. A useful exam habit is to circle the operation sign before selecting a rule.

Thinking a zero index gives zero

The student answers:

$$7^0=0$$

It feels plausible because the index is zero, but any non-zero base raised to the power zero equals $1$:

$$7^0=1$$

This follows from division:

$$\frac{7^3}{7^3}=7^{3-3}=7^0$$

But any non-zero number divided by itself is $1$, so $7^0=1$.

Treating a negative index as a negative number

A negative index does not make the value negative. It tells you to take the reciprocal:

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

A common incorrect answer is $-25$. The minus sign is part of the index, not a sign placed in front of the final value.

For a variable:

$$x^{-3}=\frac{1}{x^3}$$

If negative powers are causing confusion, the fractional and negative indices explanation connects negative indices with reciprocals in a clear GCSE example.

Reversing fractional indices

For a fractional index, the denominator gives the root and the numerator gives the power:

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

For example:

$$16^{\frac{3}{4}}=(\sqrt[4]{16}{)}^3=2^3=8$$

The student may accidentally take the cube root first because $3$ appears first in the fraction. Reading the denominator as the root prevents that mistake.

You can develop this further with the fractional and negative indices revision guide and the fractional and negative indices practice booklet.

Ignoring brackets around a negative base

Compare these expressions:

$$(-2{)}^4=16$$

and

$$-2^4=-16$$

In the first expression, the negative number is inside the brackets and is raised to the fourth power. In the second, the index applies to $2$ before the negative sign is applied.

When copying an expression into a calculator, preserve the brackets exactly as they appear in the question.

Exam-style indices practice

Try each question before reading the answer.

Simplify $a^6\times a^4$

The bases are the same and the powers are multiplied, so add the indices:

$$a^6\times a^4=a^{10}$$

Simplify $\frac{12b^7}{3b^2}$

Divide the coefficients and subtract the indices:

$$\frac{12b^7}{3b^2}=4b^{7-2}=4b^5$$

Simplify $(2x^3{)}^2$

Square the coefficient and multiply the indices:

$$(2x^3{)}^2=2^2{x}^{3\times 2}=4x^6$$

Evaluate $27^{\frac{2}{3}}$

The denominator gives the cube root:

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

Write $4x^{-2}$ using positive indices

Move the factor with the negative index into the denominator:

$$4x^{-2}=\frac{4}{x^2}$$

Notice what has changed. The student is no longer asking, “Do I add or multiply?” in isolation. They first identify the operation and the structure, then choose the corresponding law. That small pause is what protects the mark.

Turn corrected mistakes into reliable marks

Knowing the index laws is only the first step. You also need enough practice to recognise which law an exam question is testing, particularly when coefficients, brackets and several powers appear together.

Start the Maths Genie Indices Lessons and Question Bank, complete the lesson, and then use the Question Bank to correct the indices mistakes that are still costing you marks.