‹ All GCSE worksheets
Download worksheet (PDF)Answers (free account)
GCSE Maths · Higher · Chapter 16: Counting, accuracy, powers and surds · Lesson 3

Surds

Grades 6–9About 50 minutesNo calculator37 marks
Hints online, answers free with an account:
mathswariz.uk/worksheets/gcse-surds/
NameClassDate
A

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

  1. A1
    Needed today

    Work out .

    [1 mark]
    Answer
  2. A2
    Last lesson

    Work out .

    [1 mark]
    Answer
  3. A3
    From chapter 6

    The bearing of B from A is . Find the bearing of A from B.

    [1 mark]
    Answer
  4. A4
    From chapter 16

    Write as a fraction in its simplest form.

    [1 mark]
    Answer
B

Example, then your turn

Grades 6–8

Your teacher works through each example with you. Then try the one beside it.

Example 1

Simplify .

Hint Take the largest square factor out of each root first.
  1. B1

    Simplify .

    [2 marks]
    Answer
Example 2

Rationalise the denominator of .

Hint Multiply the top and the bottom by .
  1. B2

    Rationalise the denominator of .

    [3 marks]
    Answer
C

Practice

Grades 6–7
  1. C1

    Simplify .

    [2 marks]
    Answer
  2. C2

    Work out . Give your answer in its simplest form.

    [2 marks]
    Answer
  1. C3

    Expand and simplify .

    [2 marks]
    Answer
  2. C4

    Rationalise the denominator of .

    [2 marks]
    Answer
  3. C5

    Simplify .

    [2 marks]
    Answer
  4. C6

    . Find the value of .

    [2 marks]
    Answer
  5. C7

    A right-angled triangle has shorter sides of length cm and cm. The diagram is not drawn accurately. Work out the length of the longest side.

    [2 marks]
    2√3 cm√13 cm?
    Answer
  6. C8

    Expand and simplify .

    [2 marks]
    Answer
D

Exam-style questions

Grades 7–9
  1. D1

    Zoe says, "."

    Show that Zoe is wrong, and write in its simplest form.

    [2 marks]
  1. D2

    A square has sides of length cm.

    (1 + √5) cm
    (a)

    Show that the area of the square is cm.

    [2 marks]
    (b)

    Write the length of a diagonal of the square in the form .

    [2 marks]
    Answer
    (c)

    A rectangle has the same area as the square. Its width is cm. Work out its length. Give your answer in the form .

    [3 marks]
    Answer
    Total for D2: 7 marks
E

Extension

Grade 9 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

  1. E1

    Work out the exact value of .

    [3 marks]
    Hint 1 · What to try

    Rationalise the denominator of each fraction on its own.

    Hint 2 · The first line

    , so .

    Hint 3 · The full method

    In the same way the others are and . Adding, everything cancels except .

    Answer
F

How did it go?

Colour one circle on each line.
Not yetNearlyYes
I can simplify a surd and add surds with the same root.
I can multiply surds and expand brackets with surds in them.
I can rationalise the denominator of a fraction and use surds in Pythagoras.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers