Cambridge (CIE) IGCSE Your notes
Maths: Extended
Surds
Contents
Simplifying Surds
Rationalising Denominators
© 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 1
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 do calculations with surds?
Multiplying surds
You can multiply numbers under square roots together
3 × 5 = 3×5 = 15
Dividing surds
You can divide numbers under square roots
21
= 21 ÷ 7= 21 ÷7 = 3
7
Factorising surds
You can factorise numbers under square roots
35 = 5 ×7 = 5 × 7
Adding or subtracting surds
© 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 2
You can only add or subtract multiples of “like” surds
This is similar to collecting like terms when simplifying algebra Your notes
3 5+ 8 5 = 11 5
7 3 –4 3 =3 3
However 2 3 + 4 6 cannot be simplified
You cannot add or subtract numbers under square roots
Consider 9 + 4= 3 +2=5
This is not equal to 9 + 4 = 13 = 3 . 60555…
Examiner Tips and Tricks
If your calculator gives an answer as a surd, leave the value as a surd throughout the
rest of your working.
This will ensure you do not lose accuracy throughout your working.
Simplifying Surds
How do I simplify surds?
To simplify a surd, factorise the number using a square number, if possible
If multiple square numbers are a factor, use the largest
Use the fact that ab = a × b and then work out any square roots of square
numbers
E.g. 48 = 16 × 3 = 16 × 3= 4 × 3 =4 3
When simplifying multiple surds, simplify each separately
© 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 3
This may produce surds which can then be collected together
E.g. 32 + 8 can be rewritten as 16 2 + 4 2 Your notes
This simplifies to 4 2 +2 2
These surds can then be collected together
6 2
You may have to expand double brackets containing surds
This can be done in the same way as multiplying out double brackets algebraically,
and then simplifying
The property ( = a can be used to simplify the expression, once expanded
a )
2
E.g. ( 6 −2 6 + 4 expands to 6 2 + 4 6 − 2 6 − 8
) ( ) ( )
This simplifies to 6 + 2 6 − 8 which gives −2 + 2 6
Worked Example
Write 54 − 24 in the form q where q is a positive integer.
Simplify both surds separately by finding the highest square number that is a factor of
each of them
9 is a factor of 54, so 54 = 9×6 =3 6
4 is a factor of 24, so 24 = 4×6 =2 6
Simplify the whole expression by collecting the like terms
54 − 24 =3 6 −2 6 = 6
© 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 4
Rationalising Denominators
Your notes
Rationalising Denominators
What does rationalising the denominator mean?
If a fraction has a denominator containing a surd then it has an irrational denominator
4 2 2
E.g. or
3
=
5 3
The fraction can be rewritten as an equivalent fraction, but with a rational denominator
4 5 6
E.g. or
5 3
The numerator may contain a surd, but the denominator is rationalised
How do I rationalise simple denominators?
If the denominator is a surd:
Multiply the top and bottom of the fraction by the surd on the denominator
a a b
= ×
b b b
This is equivalent to multiplying by 1, so does not change the value of the
fraction
b × b = b so the denominator is no longer a surd
Multiply the fractions as you would usually, and simplify if needed
a b
b
How do I rationalise harder denominators?
If the denominator is an expression containing a surd:
2
For example
3 + 5
Multiply the top and bottom of the fraction by the expression on the denominator,
but with the sign changed
2 2 3− 5
= ×
3+ 5 3 + 5 3− 5
© 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 5
This is equivalent to multiplying by 1, so does not change the value of the
fraction
Multiply the fractions as you would usually (use brackets to help) Your notes
2 2 3− 5
=
( )
3+ 5 ( 3+ 5 3− 5 ) ( )
Expand any brackets, and simplify
2 6−2 5 6−2 5
= = 9− 5
3+ 5 ( 3×3 ) +3 5 −3 5 − ( 5 5 )
( )
You can use the difference of two squares to expand the denominator quickly
( a+ b a− b) ( ) = a2 − ( b )
2
= a2 − b
This is what makes the denominator rational
Simplify
2 6−2 5 3− 5
= =
3+ 5 4 2
Examiner Tips and Tricks
If your answer still has a surd on the bottom, go back and check your working!
Worked Example
4
Write in the form p +q r where p , q and r are integers and r has
6 −2
no square factors.
Multiply the top and bottom of the fraction by the expression on the denominator, but
with the sign changed
4 4 6 +2
= ×
6 −2 6 −2 6 +2
Multiply the fractions as you would usually
© 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 6
4 4 6 +2
=
( )
6 −2 ( 6 −2 ) ( 6 +2 )
Your notes
Expand the brackets
The denominator can be expanded using the difference of two squares
4 4 6 +8 4 6 +8
= = 6−4
6 −2 ( 6 )
2
−2 ( )
2
Simplify
4 4 6 +8
= =2 6 +4
6 −2 2
Write in the form given in the question
4+2 6
p=4
q=2
r=6
© 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 7
