Robinson Tuition
Grade 8 · Higher · free GCSE maths worksheet

Rationalising the denominator

Removing surds from fractions.

11 original questions, 42 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

x is a positive number. Show that 10√(5x) can be written as 2√(5x)x.

3 marks
Show answer
multiply top and bottom by √(5x): 10√(5x)/(5x) = 2√(5x)/x
2

Rationalise the denominator of 12√3

Give your answer in its simplest form.

2 marks
Show answer
4√3
3

Erin is asked to rationalise the denominator of 8√6

Here is her working.

8√6 = 8 × √6√6 × √6 = 8√636

Erin has made one mistake.

Explain the mistake Erin has made.

2 marks
Show answer
√6 x √6 = 6, not 36. The denominator should be 6, giving 8√66 = 4√63.
4

Write 6 + √12√3 in the form a + b√3, where a and b are integers.

4 marks
Show answer
2 + 2√3
5

x is a number greater than 4. Rationalise the denominator of 6√x − 2. Give your answer as a single fraction.

3 marks
Show answer
6(√x + 2)/(x - 4)
6

Rationalise the denominator of 224 + √5

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

4 marks
Show answer
8 - 2√5
7

The first two terms of a geometric sequence are

√5 + 1    5 + √5

Find the common ratio of the sequence.

Give your answer in its simplest form.

4 marks
Show answer
√5
8

Show that √50 + 18√2 = 14√2

5 marks
Show answer
√50 = 5√2 and 18√2 = 9√2, so the sum is 14√2
9

Show that 7 + √53 − √5 can be written as 13 + 5√52

5 marks
Show answer
Multiply top and bottom by (3 + √5): numerator (7+√5)(3+√5) = 26 + 10√5, denominator 9 - 5 = 4, so the fraction is 26+10√54 = 13+5√52
10

Express √12 + 4√3 − 1 in the form a + b√3, where a and b are integers.

Show each stage of your working.

6 marks
Show answer
5 + 3√3
11

x is a number greater than 9. Rationalise the denominator of √x + 3√x − 3. Give your answer in the form x + a√x + bx − c.

4 marks
Show answer
(x + 6√x + 9)/(x - 9)

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.

All Grade 8 topics