Surds | Pearson Edexcel GCSE Maths Notes

Visual model

Surds simplify by taking out square factors

\sqrt{72}

6\sqrt{2}

72 = 36 \times 2,\quad \sqrt{36}=6

Gold-standard guide

26 mins

What you will learn

Simplify and calculate with exact irrational values.

Use a clear step-by-step method for surds.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier number skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

A surd is an irrational root that cannot be written as an exact fraction

Step 1

Recognise the difference of two squares structure

(a + b)(a − b) = a² − b²

Step 2

Apply the formula

3² − (√5)² = 9 − 5

Step 3

State the simplified answer

9 − 5 = 4

Watch out

Students write (√5)² = √25 = 5, which accidentally gives the right answer here, but the correct reasoning is simply (√5)² = 5 because squaring undoes the square root

Simplify

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

Square factor

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

Worked example

Expand and simplify (3 + √5)(3 − √5)

Recognise the difference of two squares structure: (a + b)(a − b) = a² − b². Here a = 3 and b = √5.We use this pattern because it eliminates the surd in one step.

Apply the formula: 3² − (√5)² = 9 − 5. Remember: (√5)² = 5 because squaring a square root returns the original number.

State the simplified answer: 9 − 5 = 4.

Final answer

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Expand and simplify (3 + √5)(3 − √5)

3 marks4 minssurds-worked

Show solution

Worked solution

1.Recognise the difference of two squares structure: (a + b)(a − b) = a² − b². Here a = 3 and b = √5.We use this pattern because it eliminates the surd in one step.
2.Apply the formula: 3² − (√5)² = 9 − 5. Remember: (√5)² = 5 because squaring a square root returns the original number.
3.State the simplified answer: 9 − 5 = 4.

Final answer

Mark points

M1: recognise the difference of two squares structure
M1: apply the formula
M1: state the simplified answer
A1: 4

Watch out

Students write (√5)² = √25 = 5, which accidentally gives the right answer here, but the correct reasoning is simply (√5)² = 5 because squaring undoes the square root.More dangerously, students write (√5)² = 25, confusing squaring a square root with squaring the number inside.Always remember: (√n)² = n.

Diagnosticrecall

Simplify √48

1 mark2 minssurds-q1

Show solution

Worked solution

1.Spot the skill: A surd is an irrational root that cannot be written as an exact fraction.
2.Use the recognise the difference of two squares structure stage first, then apply the formula.
3.Keep the final answer visible: 4√3.

Final answer

4√3

Mark points

M1: use the correct a surd is an irrational root that cannot be written as an exact fraction.when you multiply a surd expression by its conjugate (same terms, opposite sign in the middle), the surds cancel out — this is called rationalising the denominator.
A1: 4√3

Watch out

Easyprocedure

Simplify √18 + √50

2 marks3 minssurds-q2

Show solution

Worked solution

1.Spot the skill: A surd is an irrational root that cannot be written as an exact fraction.
2.Use the apply the formula stage first, then state the simplified answer.
3.Keep the final answer visible: 8√2.

Final answer

8√2

Mark points

M1: use the correct a surd is an irrational root that cannot be written as an exact fraction.when you multiply a surd expression by its conjugate (same terms, opposite sign in the middle), the surds cancel out — this is called rationalising the denominator.
A1: 8√2

Watch out

Mediumreasoning

Rationalise the denominator of 6/√3, simplifying fully

3 marks4 minssurds-q3

Show solution

Worked solution

1.Spot the skill: A surd is an irrational root that cannot be written as an exact fraction.
2.Use the state the simplified answer stage first, then recognise the difference of two squares structure.
3.Keep the final answer visible: 2√3.

Final answer

2√3

Mark points

M1: use the correct a surd is an irrational root that cannot be written as an exact fraction.when you multiply a surd expression by its conjugate (same terms, opposite sign in the middle), the surds cancel out — this is called rationalising the denominator.
A1: 2√3

Watch out

Hardproblem solving

Expand and simplify (2 + √3)²

3 marks5 minssurds-q4

Show solution

Worked solution

1.Spot the skill: A surd is an irrational root that cannot be written as an exact fraction.
2.Use the recognise the difference of two squares structure stage first, then apply the formula.
3.Keep the final answer visible: 7 + 4√3.

Final answer

7 + 4√3

Mark points

M1: use the correct a surd is an irrational root that cannot be written as an exact fraction.when you multiply a surd expression by its conjugate (same terms, opposite sign in the middle), the surds cancel out — this is called rationalising the denominator.
A1: 7 + 4√3

Watch out

Exam-stylemulti-step

Show that (5 − √2)(5 + √2) = 23

4 marks6 minssurds-q5

Show solution

Worked solution

1.Spot the skill: A surd is an irrational root that cannot be written as an exact fraction.
2.Use the apply the formula stage first, then state the simplified answer.
3.Keep the final answer visible: 25 − 2 = 23 ✓.

Final answer

25 − 2 = 23 ✓

Mark points

M1: use the correct a surd is an irrational root that cannot be written as an exact fraction.when you multiply a surd expression by its conjugate (same terms, opposite sign in the middle), the surds cancel out — this is called rationalising the denominator.
A1: 25 − 2 = 23 ✓

Watch out

Grade 9 stretchproblem solving

Simplify $\sqrt{48}$ + $\sqrt{27}$ - $\sqrt{75}$ .

4 marks7 minssurd-g9

Show solution

Worked solution

1.Rewrite each surd using its largest square factor.
2.Collect the like surds.

Final answer

2sqrt(3)

Mark points

M1: obtain 4sqrt(3), 3sqrt(3), 5sqrt(3)
A1: 2sqrt(3)

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Hard exam-stylemulti-step problem

Simplify ( $\sqrt{45}$ + $\sqrt{20}$ ) / $\sqrt{5}$ .

3 marks6 minssurds-paper

Show solution

Worked solution

1.Simplify each surd in the numerator.
2.Factor out $\sqrt{5}$ .
3.Cancel the common factor.

Final answer

Mark points

M1: obtain (3sqrt(5) + 2sqrt(5)) / $\sqrt{5}$
M1: obtain 5sqrt(5) / $\sqrt{5}$
A1: obtain 5

Watch out

Read the full question before calculating. Keep each stage of your working visible.

Timed checkpoint

16 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Surds - 2 marksSimplify √48Mark answer

Answer

4√3

2Calculations and order of operations - 2 marksWork out (6 + 2) × 3 − 5Mark answer

Answer

3Integers, decimals and place value - 2 marksOrder these from smallest to largest: 0.3, 0.03, 0.303, 0.033Mark answer

Answer

0.03, 0.033, 0.3, 0.303

4Fractions - 3 marksA recipe needs ¾ cup of sugar. How much sugar is needed for 2½ batches?Mark answer

Answer

1⅞ cups

Mastery check

I can explain the method for surds.
I can show clear working without skipping key steps.
I can avoid this mistake: Students write (√5)² = √25 = 5, which accidentally gives the right answer here, but the correct reasoning is simply (√5)² = 5 because squaring undoes the square root.More dangerously, students write (√5)² = 25, confusing squaring a square root with squaring the number inside.Always remember: (√n)² = n.

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