Common Mistakes with Surds in GCSE Maths

You recognise the square root sign, remember something about square factors, and still lose marks when the expressions get longer. The common mistakes with surds in GCSE Maths usually come from a few habits: splitting roots across addition, combining unlike terms, forgetting coefficients and changing only the denominator of a fraction.

Each habit has a clear fix. Surds are irrational roots kept in exact form, and the aim is to rearrange them without changing their value. Calculating exactly with surds, simplifying them and rationalising denominators are Higher tier GCSE skills.

Imagine a student checking a marked practice sheet. Instead of writing “revise surds” at the top, they identify the first incorrect line in each answer. That gives them something specific to repair. Follow that process below, then test the fixes on fresh questions.

Your surds checking routine

Before you start, keep this checklist beside your working:

• Look for the largest square factor when simplifying.
• Simplify surds before deciding whether terms combine.
• Multiply coefficients as well as the roots.
• Expand brackets fully and keep track of signs.
• Multiply the whole fraction by an expression equal to one when rationalising.
• Keep exact answers unless the question requests an approximation.

MathsGenie's Edexcel surds topic page brings the topic's revision resources together. The AQA surds page and OCR surds page provide matching routes for those exam boards.

Common mistakes with surds in GCSE Maths

Splitting a square root across addition

The student's first incorrect line is:

$$\sqrt{9+16}=\sqrt{9}+\sqrt{16}=7.$$

Why it happens: a useful multiplication rule gets applied to addition. For non-negative numbers, roots can be separated across multiplication:

$$\sqrt{ab}=\sqrt{a}\times \sqrt{b}.$$

There is no equivalent general rule for splitting a root across a sum.

Correct method: calculate the expression under the root first:

$$\sqrt{9+16}=\sqrt{25}=5.$$

The root sign groups everything beneath it. Treat that expression as a single quantity before taking its square root.

Stopping before a surd is fully simplified

Next, the student writes:

$$\sqrt{72}=\sqrt{4\times 18}=2\sqrt{18}.$$

This line is correct, but the answer is unfinished because the remaining root still contains a square factor.

Why it happens: finding any square factor feels like completing the task.

Correct method: use the largest square factor you can identify:

$$\sqrt{72}=\sqrt{36\times 2}=6\sqrt{2}.$$

The earlier route also works if you continue:

$$2\sqrt{18}=2\sqrt{9\times 2}=2\times 3\sqrt{2}=6\sqrt{2}.$$

Check the number left under the root. If it has a square factor greater than one, there is more simplifying to do.

Adding the numbers inside the roots

Another answer reads:

$$\sqrt{12}+\sqrt{27}=\sqrt{39}.$$

Why it happens: the student treats two roots as though they can be merged across an addition sign.

Correct method: simplify each term separately:

$$\sqrt{12}=2\sqrt{3},\sqrt{27}=3\sqrt{3}.$$

Now the terms have the same root part, so their coefficients combine:

$$2\sqrt{3}+3\sqrt{3}=5\sqrt{3}.$$

Think of this as collecting like terms in algebra. Two lots of a quantity plus three lots make five lots. If the simplified root parts differ, leave the terms separate. For example, $\sqrt{2}+\sqrt{3}$ cannot be collected into one surd term.

Forgetting the coefficients when multiplying

The student correctly multiplies the roots but loses the numbers in front:

$$(2\sqrt{3})(4\sqrt{6})=\sqrt{18}.$$

Why it happens: attention goes straight to the unfamiliar symbols, while ordinary multiplication gets overlooked.

Correct method: multiply the coefficients and the roots, then simplify:

$$(2\sqrt{3})(4\sqrt{6})=8\sqrt{18}=8\times 3\sqrt{2}=24\sqrt{2}.$$

Writing the coefficient multiplication explicitly makes missing factors easier to spot. The surds revision guide gives you a place to revisit the topic's methods before attempting more questions.

Squaring only the separate terms in a bracket

The next mistake looks plausible:

$$(3+\sqrt{2}{)}^2=9+2=11.$$

Why it happens: squaring a bracket is confused with squaring each term independently. The two cross-products disappear.

Correct method: write the square as two identical brackets and multiply every term:

$$(3+\sqrt{2})(3+\sqrt{2})=9+3\sqrt{2}+3\sqrt{2}+2=11+6\sqrt{2}.$$

Keeping both middle terms visible helps prevent this error. It also makes the working easier to check than jumping straight to an answer.

Changing only the denominator when rationalising

Finally, the student tries to remove a root from a denominator:

$$\frac{5}{\sqrt{3}}=\frac{5}{3}.$$

Why it happens: “remove the root” becomes the whole goal, and preserving the fraction's value gets forgotten.

Correct method: multiply the numerator and denominator by the same non-zero expression:

$$\frac{5}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{5\sqrt{3}}{3}.$$

Because $\frac{\sqrt{3}}{\sqrt{3}}=1$, the value stays unchanged.

For a denominator containing two terms, use its conjugate: the same terms with the sign between them changed. For example:

$$\frac{1}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}}=\frac{2-\sqrt{3}}{4-3}=2-\sqrt{3}.$$

The denominator becomes rational because the middle terms cancel. Multiply out the numerator as well, and simplify the final fraction where possible.

Start with MathsGenie's Surds Lessons and Question Bank to revisit the method behind your first incorrect line, then practise it through the topic's Question Bank.

A stick-figure student asks for one tiny correction while a hard-hat-wearing eraser discovers a mountain of recurring mistakes.

Exam-style practice: test the repairs

Try these without looking back. Show enough working that someone else could follow your method.

• Simplify $\sqrt{108}$.
• Simplify $\sqrt{45}+\sqrt{20}$.
• Simplify $(3\sqrt{2})(2\sqrt{6})$.
• Expand and simplify $(4-\sqrt{3}{)}^2$.
• Rationalise the denominator of $\frac{7}{\sqrt{5}}$.
• Rationalise the denominator of $\frac{2}{3+\sqrt{7}}$.

Answers and checks

First question:

$$\sqrt{108}=\sqrt{36\times 3}=6\sqrt{3}.$$

Second question:

$$\sqrt{45}+\sqrt{20}=3\sqrt{5}+2\sqrt{5}=5\sqrt{5}.$$

Third question:

$$(3\sqrt{2})(2\sqrt{6})=6\sqrt{12}=12\sqrt{3}.$$

Fourth question:

$$(4-\sqrt{3}{)}^2=16-4\sqrt{3}-4\sqrt{3}+3=19-8\sqrt{3}.$$

Fifth question:

$$\frac{7}{\sqrt{5}}=\frac{7\sqrt{5}}{5}.$$

Sixth question:

$$\frac{2}{3+\sqrt{7}}=\frac{2(3-\sqrt{7})}{9-7}=3-\sqrt{7}.$$

For each wrong answer, underline the first incorrect line. Then write one instruction beside it, such as “multiply both coefficients” or “expand all four products”. Redo the question without copying the solution, and try another of the same type later.

Keep surds exact throughout your working. Replacing $\sqrt{2}$ with a rounded decimal changes an exact expression into an approximation and can prevent you giving the requested answer.

Turn corrections into reliable working

Our imagined student's revision now has a clear purpose: repair one recurring error, then check whether the method holds on a fresh question. You can use the same approach to close the gap between recognising a solution and producing it independently. Start your surds revision with MathsGenie's Lessons and Question Bank, focusing first on the mistake you found in your own working.