GCSE Maths · Topic guide
Surds
A surd is a root, usually a square root, that cannot be simplified to a whole number, such as the square root of 2 or the square root of 7. Because their decimal values go on forever without repeating, surds are left in root form to keep answers exact rather than rounding them.
Method
- To simplify a surd, look for a square number factor (4, 9, 16, 25...) and split it out: sqrt(ab) = sqrt(a) x sqrt(b).
- To add or subtract surds, they must have the same number under the root sign; simplify first, then combine like terms.
- To multiply surds, multiply the numbers under the root signs together: sqrt(a) x sqrt(b) = sqrt(ab).
- To rationalise a denominator with a single surd, multiply top and bottom by that surd so the root disappears from the bottom.
- To rationalise a denominator such as 1/(a + sqrt(b)), multiply top and bottom by the conjugate (a - sqrt(b)), using the difference-of-two-squares result to remove the surd.
Worked example
Simplify sqrt(48), and then rationalise 1/sqrt(48) fully.
- 48 = 16 x 3, and 16 is a square number, so sqrt(48) = sqrt(16) x sqrt(3) = 4sqrt(3).
- 1/sqrt(48) is the same as 1/(4sqrt(3)).
- Multiply top and bottom by sqrt(3): (1 x sqrt(3)) / (4sqrt(3) x sqrt(3)) = sqrt(3) / (4 x 3).
- sqrt(3) x sqrt(3) = 3, so the denominator becomes 12.
- The fully rationalised answer is sqrt(3)/12.
Practice questions
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Exam-style questions
Written in the style of a GCSE exam paper, with a full mark scheme.
Q1[3 marks]
Show that (3 + sqrt(2))^2 can be written in the form a + b sqrt(2), where a and b are integers.
Q2[3 marks]
Rationalise the denominator of 4/(1 + sqrt(3)). Give your answer in its simplest form.
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