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Surds - Teacher Slides.pptx
The lesson slides with the teacher's notes on each slide, and every question and mark scheme built in. Built from the lesson script on 28 September 2026.
EDEXCEL GCSE MATHS · HIGHER TIER
Surds
Number
Lesson 7 of 7
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EDEXCEL GCSE MATHS · HIGHER TIER
Last Lesson and Before
Number
Lesson 7 of 7
Answer each one, then check.
1
What is √36?
2
What is √4 × √9?
3
Expand 3(x + 2).
4
Last lesson: write 4 × 10³ as an ordinary number.
5
Which of these is a whole number: √16 or √17?
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EDEXCEL GCSE MATHS · HIGHER TIER
Last Lesson and Before - Answers
Number
Lesson 7 of 7
1
What is √36?
6
2
What is √4 × √9?
2 × 3 = 6 - the same as √36.
3
Expand 3(x + 2).
3x + 6
4
Last lesson: write 4 × 10³ as an ordinary number.
4000
5
Which of these is a whole number: √16 or √17?
√16 = 4. √17 is not a whole number.
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EDEXCEL GCSE MATHS · HIGHER TIER
Learning Objectives
Number
Lesson 7 of 7
1
Know what rational and irrational numbers are, and what a surd is.
2
Simplify surds such as √72.
3
Multiply, divide, add and subtract surds.
4
Expand brackets containing surds.
5
Rationalise the denominator of a fraction such as 6/(√3).
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EDEXCEL GCSE MATHS · HIGHER TIER
What Is a Surd?
Number
Lesson 7 of 7
A surd is a root that cannot be written exactly as a fraction or a terminating or recurring decimal.
Rational numbers
Can be written as a fraction of two integers: ¾, −7, 0.6, √25 = 5.
Irrational numbers
Cannot: √2 = 1.414 213 56…, π= 3.141 592 65… The decimals never end and never repeat.
Surds
Roots that are irrational: √2, √10, ∛5. √9 is not a surd, because √9 = 3.
Why use them
√2 is exact; 1.414 is not. When a question says "give your answer in surd form", it wants the exact answer.
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EDEXCEL GCSE MATHS · HIGHER TIER
Rational or Irrational?
Number
Lesson 7 of 7
RATIONAL
• ¾
• 0.666…= ⅔
• √25 = 5
• −7
• √(4/9) = ⅔
IRRATIONAL
• √2
• √10
• π
• ∛5
• 1 + √3
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EDEXCEL GCSE MATHS · HIGHER TIER
The Rules for Surds
Number
Lesson 7 of 7
Roots can be split up and combined when multiplying and dividing - but not when adding.
Multiplying
√a × √b = √ab. √2 × √8 = √16 = 4.
Dividing
√a/(√b) = √(a/b). √50/(√2) = √25 = 5.
A surd times itself
√a × √a = a. √7 × √7 = 7.
Not for adding
√(a + b) is NOT √a + √b: √(9 + 16) = √25 = 5, but √9 + √16 = 7.
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EDEXCEL GCSE MATHS · HIGHER TIER
Surds in Right-Angled Triangles
Number
Lesson 7 of 7

The long side of a right-angled triangle is often a surd.
Surds turn up all the time in geometry. A right-angled triangle with shorter sides 1 and 1 has a hypotenuse of exactly √2 - a length you can draw perfectly but never write exactly as a decimal.
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EDEXCEL GCSE MATHS · HIGHER TIER
PART ONE
Simplifying Surds
Take out the biggest square factor.
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EDEXCEL GCSE MATHS · HIGHER TIER
Simplifying a Surd
Number
Lesson 7 of 7
Simplify √72
1
Find the largest square number that is a factor of 72
Square numbers: 4, 9, 16, 25, 36 ... and 36 × 2 = 72
2
Split the root
√72 = √36 × √2
3
Work out the square root
√36 = 6
ANSWER
√72 = 6√2
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EDEXCEL GCSE MATHS · HIGHER TIER
Adding Surds
Number
Lesson 7 of 7
Simplify √50 + √18
1
Simplify each surd
√50 = √25 × √2 = 5√2 and √18 = √9 × √2 = 3√2
2
They are now "like surds" - both multiples of √2
5√2 + 3√2
3
Add them like algebra: 5x + 3x = 8x
8√2
ANSWER
8√2
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EDEXCEL GCSE MATHS · HIGHER TIER
Expanding Brackets with Surds
Number
Lesson 7 of 7
Expand and simplify (3 + √2)(5 − √2)
1
First times first
3 × 5 = 15
2
Outer and inner
3 × (−√2) = −3√2 and √2 × 5 = 5√2
3
Last times last
√2 × (−√2) = −2
4
Collect like terms
15 − 2 = 13 and −3√2 + 5√2 = 2√2
ANSWER
13 + 2√2
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EDEXCEL GCSE MATHS · HIGHER TIER
PART TWO
Rationalising the Denominator
Getting the surd out of the bottom of a fraction.
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EDEXCEL GCSE MATHS · HIGHER TIER
Rationalising the Denominator
Number
Lesson 7 of 7
To rationalise a fraction like a/(√b), multiply the top AND the bottom by √b.
Why it works
Multiplying top and bottom by the same number does not change a fraction's value, and √b × √b = b removes the surd from the bottom.
Example
6/(√3) = (6 × √3)/(√3 × √3) = 6√3/3 = 2√3.
Always simplify
Cancel any common factors, and simplify any surd left on top.
Exam wording
"Rationalise the denominator" and "give your answer in the form a√b" both mean do this.
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EDEXCEL GCSE MATHS · HIGHER TIER
Rationalising
Number
Lesson 7 of 7
Rationalise the denominator of 10/(√5). Give your answer in its simplest form.
1
Multiply the top and the bottom by √5
(10 × √5)/(√5 × √5)
2
The bottom becomes a whole number
√5 × √5 = 5
3
So
10√5/5
4
Simplify
10 ÷ 5 = 2
ANSWER
2√5
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EDEXCEL GCSE MATHS · HIGHER TIER
Number
Lesson 7 of 7
CASE STUDY
A4 Paper and the Square Root of 2
Every A-size sheet of paper - A3, A4, A5 - has its long side √2 times its short side. That is the one shape that stays exactly the same when you fold it in half: halve the long side of a sheet with sides 1 and √2 and you get sides √2/2 and 1, which is the same shape again. It is why an A4 page shrinks perfectly onto A5 on a photocopier. An A4 sheet is 210 mm by 297 mm, and 297 ÷ 210 = 1.414…, which is √2 to the nearest millimetre.
210 × 297 mm
The size of an A4 sheet
1 : √2
The ratio of the sides of every A-size sheet
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EDEXCEL GCSE MATHS · HIGHER TIER
Folding A-Size Paper
Number
Lesson 7 of 7

Fold A3 in half and you get A4; fold A4 in half and you get A5.
Halving an A-size sheet always makes the next size down, with exactly the same shape. The ratio of the sides is 1 : √2 at every size - a surd you use every day.
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EDEXCEL GCSE MATHS · HIGHER TIER
Key Terms
Number
Lesson 7 of 7
Rational number
A number that can be written as a fraction of two integers.
Irrational number
A number that cannot be written as a fraction; its decimal never ends or repeats.
Surd
A root that is irrational, such as √2 or √10.
Like surds
Multiples of the same surd, such as 5√2 and 3√2, which can be added and subtracted.
Simplest form
A surd with the largest possible square number taken out, e.g. 6√2.
Rationalise the denominator
Rewrite a fraction so there is no surd on the bottom.
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EDEXCEL GCSE MATHS · HIGHER TIER
Number
Lesson 7 of 7
YOUR TASK
Surd Simplifying Race
12 minutes
Simplify each. (a) √12 (b) √45 (c) √200 (d) √8 + √32 (e) √75 − √27 (f) √3 × √15 (g) (1 + √3)² (h) 12/(√6)
1
Look for the largest square factor.
2
Simplify before adding or subtracting.
3
Rationalise any surd in the denominator.
WHAT A GOOD ANSWER SHOWS
(a) 2√3 (b) 3√5 (c) 10√2 (d) 2√2 + 4√2 = 6√2 (e) 5√3 − 3√3 = 2√3 (f) √45 = 3√5 (g) 1 + 2√3 + 3 = 4 + 2√3 (h) 12√6/6 = 2√6
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EDEXCEL GCSE MATHS · HIGHER TIER
Can I...?
Number
Lesson 7 of 7
Explain what a surd is.
Say whether a number is rational or irrational.
Simplify a surd such as √72.
Multiply and divide surds.
Add and subtract like surds.
Expand brackets containing surds.
Rationalise a denominator like a/(√b).
Give an exact answer in surd form.
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EDEXCEL GCSE MATHS · HIGHER TIER
Summary & Exam Focus
A surd is an irrational root, and gives an exact value.
√ab = √a × √b: take out the largest square factor to simplify.
Only like surds can be added or subtracted.
Rationalise a/(√b) by multiplying the top and the bottom by √b.
EXAM FOCUS
Rationalise the denominator of 12/(√3). Give your answer in its simplest form. (2 marks)
"Exact" or "surd form" means do not use a decimal. Leave the root in the answer and simplify it as far as it goes.
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EDEXCEL GCSE MATHS · HIGHER TIER
Exam Practice: Surds
Number
Lesson 7 of 7
Answer all questions. Show your working. All questions are non-calculator. · 25 minutes
Question 1 · 1 mark · Non-calculator
Write √45 in the form k√5, where k is an integer.
Question 2 · 2 marks · Non-calculator
Rationalise the denominator of 12/(√3). Give your answer in its simplest form.
Question 3 · 3 marks · Non-calculator
Expand and simplify (2 + √3)(4 − √3). Give your answer in the form a + b√3, where a and b are integers.
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EDEXCEL GCSE MATHS · HIGHER TIER
Exam Practice: Surds (cont.)
Number
Lesson 7 of 7
Question 4 · 3 marks · Non-calculator · Show that
Show that √98 + √8 = 9√2
Question 5 · 3 marks · Non-calculator
The diagram shows the right-angled triangle ABC. AB = √5 cm and BC = √11 cm. Work out the length of AC.
Question 6 · 3 marks · Non-calculator
A rectangle has length (3 + √5) cm and width (3 − √5) cm. Work out the area of the rectangle.
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EDEXCEL GCSE MATHS · HIGHER TIER
Question 1 · 1 mark · Non-calculator
Number
Lesson 7 of 7
“
Write √45 in the form k√5, where k is an integer.
HOW TO ANSWER IT
Command word: Non-calculator. Worth 1 mark, so plan before writing.
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EDEXCEL GCSE MATHS · HIGHER TIER
Question 1 · mark scheme
Number
Lesson 7 of 7
1 mark available. Award a mark for each point made.
3√5
B1
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EDEXCEL GCSE MATHS · HIGHER TIER
Question 2 · 2 marks · Non-calculator
Number
Lesson 7 of 7
“
Rationalise the denominator of 12/(√3). Give your answer in its simplest form.
HOW TO ANSWER IT
Command word: Non-calculator. Worth 2 marks, so plan before writing.
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EDEXCEL GCSE MATHS · HIGHER TIER
Question 2 · mark scheme
Number
Lesson 7 of 7
2 marks available. Award a mark for each point made.
Multiplying the top and the bottom by √3
M1
4√3
A1
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EDEXCEL GCSE MATHS · HIGHER TIER
Question 3 · 3 marks · Non-calculator
Number
Lesson 7 of 7
“
Expand and simplify (2 + √3)(4 − √3). Give your answer in the form a + b√3, where a and b are integers.
HOW TO ANSWER IT
Command word: Non-calculator. Worth 3 marks, so plan before writing.
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EDEXCEL GCSE MATHS · HIGHER TIER
Question 3 · mark scheme
Number
Lesson 7 of 7
3 marks available. Award a mark for each point made.
At least 3 of the 4 terms correct: 8, −2√3, 4√3, −3
M1
All 4 terms correct
M1
5 + 2√3
A1
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EDEXCEL GCSE MATHS · HIGHER TIER
Question 4 · 3 marks · Non-calculator · Show that
Number
Lesson 7 of 7
“
Show that √98 + √8 = 9√2
HOW TO ANSWER IT
Command word: Non-calculator · Show that. Worth 3 marks, so plan before writing.
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EDEXCEL GCSE MATHS · HIGHER TIER
Question 4 · mark scheme
Number
Lesson 7 of 7
3 marks available. Award a mark for each point made.
√98 = 7√2 or √8 = 2√2
M1
Both simplified correctly
M1
Correctly added to reach 9√2
A1
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EDEXCEL GCSE MATHS · HIGHER TIER
Question 5 · 3 marks · Non-calculator
Number
Lesson 7 of 7

The diagram shows the right-angled triangle ABC. AB = √5 cm and BC = √11 cm. Work out the length of AC. (3 marks)
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EDEXCEL GCSE MATHS · HIGHER TIER
Question 5 · mark scheme
Number
Lesson 7 of 7
3 marks available. Award a mark for each point made.
(√5)² + (√11)² or 5 + 11
M1
√16
M1
4 (cm)
A1
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EDEXCEL GCSE MATHS · HIGHER TIER
Question 6 · 3 marks · Non-calculator
Number
Lesson 7 of 7
“
A rectangle has length (3 + √5) cm and width (3 − √5) cm. Work out the area of the rectangle.
HOW TO ANSWER IT
Command word: Non-calculator. Worth 3 marks, so plan before writing.
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EDEXCEL GCSE MATHS · HIGHER TIER
Question 6 · mark scheme
Number
Lesson 7 of 7
3 marks available. Award a mark for each point made.
Area = (3 + √5)(3 − √5)
M1
At least 3 of the 4 terms correct: 9, −3√5, +3√5, −5
M1
4 (cm²)
A1
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