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Cambridge O Level Maths Your notes
Surds
Contents
Simplifying Surds
Rationalising Denominators
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Simplifying Surds
Your notes
Surds & Exact Values
What is a surd?
A surd is the square root of a non-square integer
Using surds lets you leave answers in exact form
e.g. 5 2 rather than 7 . 071067812
How do I calculate with surds?
Multiplying surds
You can multiply numbers under square roots together
eg. 3 × 5 = 3 × 5 = 15
Dividing surds
You can divide numbers under square roots
eg. 21 ÷ 7 = 21 ÷ 7 = 3
Factorising surds
You can factorise numbers under square roots
eg. 35 = 5 × 7 = 5 × 7
Adding or subtracting surds is very like adding or subtracting letters in algebra – you can only add or
subtract multiples of “like” surds
eg. 3 5 + 8 5 = 11 5 or 7 3 – 4 3 = 3 3
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Be very careful here, you can not add or subtract numbers under square roots
Think about 9 + 4= 3 + 2 = 5 Your notes
It is not equal to 9 + 4 = 13 = 3 . 60555 …
Examiner Tip
If you are working on an exam question and your calculator gives you an answer as a surd, leave the
value as a surd throughout the rest of your calculations to make sure you do not lose accuracy
throughout your questions
Page 3 of 6
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Rationalising Denominators
Your notes
Rationalising Denominators
What does it mean to rationalise a denominator?
If a fraction has a surd on the denominator, it is not in its simplest form and must be rationalised
Rationalising a denominator changes a fraction with surds in the denominator into an equivalent
fraction
The denominator will be an integer and any surds are in the numerator
How do I rationalise the denominator of a surd?
To rationalise the denominator if the denominator is a surd
STEP 1: Multiply the top and bottom by the surd on the denominator:
a a b
= ×
b b b
This ensures we are multiplying by 1; so not affecting the overall value
STEP 2: Multiply the numerator and denominators together
b × b = b so the denominator is no longer a surd
STEP 3: Simplify your answer if needed
To rationalise the denominator if the denominator is an expression containing a surd:
2
For example
1+ 3
STEP 1: Multiply the top and bottom by the expression on the denominator, but with the sign
changed
2 1− 3
×
1+ 3 1− 3
This ensures we are multiplying by 1; so not affecting the overall value
STEP 2: Multiply the expressions on the numerator and denominator together
(a + b ) (a − b ) = a 2 + a b − a b − b = a 2 − b so the
denominator no longer contains a surd
STEP 3: Simplify your answer if needed
2(1 − 3) 2(1 − 3 ) 2(1 − 3)
= = − = − (1 − 3) = 3 −1
(1 + 3 ) (1 − 3) 1−3 2
Page 4 of 6
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Examiner Tip
Your notes
When you have an expression on the denominator you can use the FOIL technique from
multiplying out double brackets
Remember that the aim is to remove the surd from the denominator, so if this doesn't happen
you need to check your working or rethink the expression you are using in your calculation
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Worked example
Your notes
4
Write in the form p + q r where p , q and r are integers and r has no square factors.
6 −2
There is an expression on the denominator, so the fraction will need to be multiplied by a fraction with
this expression on both the numerator and denominator, but with the sign changed.
Multiply the fractions together by multiplying across the numerator and the denominator.
By expanding the denominator, you will notice that it is a difference of two squares problem.
Simplify by cancelling out the 4 on the numerator and the 2 on the denominator.
Expand and write in the form given in the question.
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