Are surds on the Foundation paper?

Not the manipulation. Foundation students meet spec ref N8 only as calculating exactly with fractions and multiples of ; simplifying surd expressions and rationalising denominators are marked as Higher tier content in 1MA1. If you are entered for Foundation you will not be asked to write sqrt(12) as 2sqrt(3).

How many marks are surds worth on Edexcel Higher?

Usually 2-5 marks per paper. Simplifying a single root is 1-2 marks, expanding brackets containing surds is 2-3, and rationalising a denominator of the form a + sqrt(b) is typically 3. Surds also appear inside Pythagoras, area and volume questions where the exact answer is required, so the real total across a series is higher than the questions labelled as surd questions.

Do surds only come up on the non-calculator paper?

No. Edexcel can set them on Paper 1, Paper 2 or Paper 3. They are most common on Paper 1 because it is non-calculator, but a calculator does not help on Papers 2 and 3 when the answer has to be exact: a question asking for the form asqrt(b) will not accept 8.66.

Do I have to rationalise the denominator if the question does not say so?

If the question says "in its simplest form" or gives you a form to write the answer in, then yes, and the final accuracy mark depends on it. When neither phrase appears, a fraction with a surd underneath is usually still accepted, but rationalising costs one line and removes the risk.

Surds - Edexcel GCSE Higher Maths

Typically 2-5 marks per paper, concentrated on Paper 1 · Spec 1MA1

A surd is a root with no exact decimal, so it stays in root form: $\sqrt{2}$ and $\sqrt{12}$ are surds, $\sqrt{9}$ is not. On Edexcel's 1MA1 specification this is spec ref N8, and the surd half of that reference (simplifying expressions such as $\sqrt{12} = 2\sqrt{3}$ , and rationalising denominators) is Higher tier only. Surds sit under the algebra reference A4 as well, which is how a pair of brackets like $(3\sqrt{2} - 1)(2\sqrt{2} + 5)$ ends up in an expand-and-simplify question.

Surds can appear on any of the three papers, but they cluster on Paper 1, which is non-calculator. On Papers 2 and 3 a calculator buys you nothing: once the question says "give your answer in the form $a\sqrt{b}$ " or "in its simplest form", a decimal scores zero however accurate it is. The question shapes repeat from series to series. Simplifying one root is 1-2 marks near the front of the paper, expanding a pair of brackets is 2-3 marks in the middle, and rationalising a two-term denominator using the conjugate is a 3-mark question late on, sometimes disguised as the hypotenuse of a right-angled triangle or the area of a rectangle.

The marks disappear in the same handful of places. Writing $\sqrt{3} \times \sqrt{3}$ as 9 ruins a rationalising question at the last step, and taking the square factor instead of its root turns $\sqrt{50}$ into $25\sqrt{2}$ . Method marks survive arithmetic slips, so write the conjugate line down before you simplify anything: multiplying numerator and denominator by $3 - \sqrt{5}$ earns its M mark even if the expansion afterwards goes wrong. Practise until $\sqrt{a} \times \sqrt{a} = a$ is automatic, then drill the conjugate until you stop copying the sign across unchanged.

Worked example

Write

\frac{12}{3 - \sqrt{3}}

in the form

a + b\sqrt{3}

, where

a

and

b

are integers.

[4 marks]

Multiply the numerator and the denominator by the conjugate of the denominator, which is $3 + \sqrt{3}$ : $\frac{12}{3 - \sqrt{3}} \times \frac{3 + \sqrt{3}}{3 + \sqrt{3}}$
M1: multiplying top and bottom by $3 + \sqrt{3}$ , the same two terms with the sign between them reversed
Expand the denominator. It is a difference of two squares, so the surd terms cancel: $(3 - \sqrt{3})(3 + \sqrt{3}) = 9 + 3\sqrt{3} - 3\sqrt{3} - 3 = 6$ .
M1: a rational denominator, using $\sqrt{3} \times \sqrt{3} = 3$ and not 9
Expand the numerator: $12(3 + \sqrt{3}) = 36 + 12\sqrt{3}$ .
M1: both terms in the bracket multiplied by 12
Divide every term by 6: $\frac{36 + 12\sqrt{3}}{6} = 6 + 2\sqrt{3}$ , so $a = 6$ and $b = 2$ .
A1: fully simplified and in the form the question asked for

Practice questions

These are original questions written in Edexcel 1MA1 Higher style, from single-root simplifying up to conjugate rationalising. Work them without a calculator, as you would on Paper 1. Where a question asks for the form $a\sqrt{b}$ or $a + b\sqrt{c}$ , type the integers into the labelled boxes, and remember that $b$ must have no square factors left.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

Question 1 Warm-up [2 marks]

Simplify $3\sqrt{5} + 4\sqrt{5}$ . Write your answer in the form $a\sqrt{b}$ , where $b$ is as small as possible, and give the values of $a$ and $b$ .

a = b =
Question 2 Exam pace [3 marks]

Simplify $\sqrt{50} + \sqrt{18} - \sqrt{8}$ . Write your answer in the form $a\sqrt{b}$ , where $b$ is as small as possible, and give the values of $a$ and $b$ .

a = b =
Question 3 Exam pace [1 mark]

Which one of these surds is already fully simplified?

Answer options for question 3
$2\sqrt{8}$ $3\sqrt{12}$ $5\sqrt{6}$ $4\sqrt{18}$
Question 4 Stretch [3 marks]

Rationalise the denominator of $\frac{8}{3 - \sqrt{7}}$ . Write your answer in the form $a + b\sqrt{7}$ , where $a$ and $b$ are integers, and give the values of $a$ and $b$ .

a = b =
Question 5 Stretch [3 marks]

Rationalise the denominator of $\frac{15}{2\sqrt{3}}$ . Write your answer in the form $\frac{a\sqrt{3}}{b}$ , where $a$ and $b$ are integers with no common factor, and give the values of $a$ and $b$ .

a = b =
Question 6 Stretch [3 marks]

A square has a perimeter of $8\sqrt{2}$ cm. Work out the area of the square, in square centimetres.

Your answer (fractions like 1/2 are fine)

Common mistakes examiners see

Writing the square factor instead of its root, so $\sqrt{50} = \sqrt{25 \times 2}$ becomes $25\sqrt{2}$ rather than $5\sqrt{2}$ .

Split the number as a square factor times the rest, then take the root of the square factor as it moves outside: $\sqrt{25} = 5$ . Finish by checking that what is left underneath has no square factors of its own, or a 2-mark question will only score 1.
Multiplying a root by itself and squaring the number underneath, giving $\sqrt{3} \times \sqrt{3} = 9$ or $\sqrt{5} \times \sqrt{5} = 25$ .

Learn $\sqrt{a} \times \sqrt{a} = a$ as a fact in its own right and say it out loud before you use it. It is the step that turns a correct rationalising method into a wrong final answer, so check it every time the denominator comes out.
Adding the numbers under the roots, so $\sqrt{9} + \sqrt{16}$ is given as $\sqrt{25} = 5$ instead of $3 + 4 = 7$ .

Roots only combine over multiplication and division, never over addition. Simplify each surd first, then add only the terms with the same root: $\sqrt{27} + \sqrt{12} = 3\sqrt{3} + 2\sqrt{3} = 5\sqrt{3}$ , while $2\sqrt{3} + 4\sqrt{5}$ cannot be shortened at all.
Expanding a bracket and leaving a root behind, so $\sqrt{3}(2\sqrt{3} - 4)$ is given as $2\sqrt{3} - 4\sqrt{3}$ because $\sqrt{3} \times 2\sqrt{3}$ was written as $2\sqrt{3}$ rather than 6.

Multiply the outside term into each term inside separately and write both products before combining. For two brackets, count four products before you simplify; on a squared bracket like $(\sqrt{7} + 2)^2$ that means writing it out as $(\sqrt{7} + 2)(\sqrt{7} + 2)$ rather than squaring the two terms.
Rationalising $\frac{5}{2 + \sqrt{3}}$ by multiplying by $\frac{2 + \sqrt{3}}{2 + \sqrt{3}}$ , or by multiplying the denominator only, so a surd is still on the bottom.

Change the sign between the two terms to get the conjugate, then multiply numerator and denominator by it so the fraction keeps its value. Check the finished answer: if a root is still underneath, the accuracy mark has gone, so go back to the expansion of the denominator.

Frequently asked questions

Are surds on the Foundation paper?: Not the manipulation. Foundation students meet spec ref N8 only as calculating exactly with fractions and multiples of $\pi$ ; simplifying surd expressions and rationalising denominators are marked as Higher tier content in 1MA1. If you are entered for Foundation you will not be asked to write $\sqrt{12}$ as $2\sqrt{3}$ .
How many marks are surds worth on Edexcel Higher?: Usually 2-5 marks per paper. Simplifying a single root is 1-2 marks, expanding brackets containing surds is 2-3, and rationalising a denominator of the form $a + \sqrt{b}$ is typically 3. Surds also appear inside Pythagoras, area and volume questions where the exact answer is required, so the real total across a series is higher than the questions labelled as surd questions.
Do surds only come up on the non-calculator paper?: No. Edexcel can set them on Paper 1, Paper 2 or Paper 3. They are most common on Paper 1 because it is non-calculator, but a calculator does not help on Papers 2 and 3 when the answer has to be exact: a question asking for the form $a\sqrt{b}$ will not accept 8.66.
Do I have to rationalise the denominator if the question does not say so?: If the question says "in its simplest form" or gives you a form to write the answer in, then yes, and the final accuracy mark depends on it. When neither phrase appears, a fraction with a surd underneath is usually still accepted, but rationalising costs one line and removes the risk.

Surds - Edexcel GCSE Higher Maths

Worked example

Practice questions

Common mistakes examiners see

Frequently asked questions

Related topics