Robinson Tuition
Grade 8 · Higher · free GCSE maths worksheet

Expanding and simplifying surd expressions

Multiplying brackets containing roots.

10 original questions, 35 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Expand and simplify (6 + √11)(6 − √11)

2 marks
Show answer
25
2

Expand and simplify (5 + 2√3)(5 − 2√3)

2 marks
Show answer
13
3

Expand and simplify (2 + √5)(3 + √5)

3 marks
Show answer
11 + 5√5
4

Expand and simplify (2 + 3√2)(4 − √2)

Give your answer in the form a + b√2, where a and b are integers.

3 marks
Show answer
2 + 10√2
5

Expand and simplify (3 − 2√5)2

Give your answer in the form a + b√5, where a and b are integers.

3 marks
Show answer
29 - 12√5
6

Expand and simplify (4 − √3)2

Give your answer in the form a + b√3, where a and b are integers.

3 marks
Show answer
19 - 8√3
7

Show that (2 + √3)2 + (2 − √3)2 = 14

4 marks
Show answer
shown: 7 + 4√3 + 7 - 4√3 = 14
8

Show that (√6 + √2)2 = 8 + 4√3

4 marks
Show answer
shown: 6 + 2√12 + 2 = 8 + 4√3
9

A right-angled triangle has perpendicular sides of length (3 + √2) cm and (3 − √2) cm.

  1. Show that the area of the triangle is 3.5 cm2.
  2. Find the exact length of the hypotenuse.
5 marks
Show answer
a) shown: 1/2 x 7 = 3.5 b) √22 cm
10

A trapezium has parallel sides of length (3 + √2) cm and (5 + 3√2) cm.

The perpendicular distance between the parallel sides is (4 − √2) cm.

Work out the exact area of the trapezium.

Give your answer in the form (a + b√2) cm², where a and b are integers.

(5 + 3√2) cm(3 + √2) cm(4 − √2) cmDiagram NOT accurately drawn
6 marks
Show answer
12 + 4√2

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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