Surds are numbers left in 'square root form' (or
'cube root form' etc). They are therefore irrational numbers. The reason we
leave them as surds is because in decimal form they would go on forever and so
this is a very clumsy way of writing them.
Addition and Subtraction of Surds
Adding and subtracting surds are simple however we need the
numbers being square rooted (or cube rooted etc) to be the same.
4Ö7  2Ö7 = 2Ö7. 5Ö2
+ 8Ö2 = 13Ö2
Note: 5Ö2 +
3Ö3 cannot be manipulated because the
surds are different (one is Ö2 and one
is Ö3).
Multiplication
Ö5 × Ö15 = Ö75 (= 15 × 5) = Ö25 × Ö3 = 5Ö3.
(1 + Ö3) × (2  Ö8) [The
brackets are expanded as usual] = 2  Ö8 + 2Ö3  Ö24 = 2  2Ö2 + 2Ö3  2Ö6
Rationalising the Denominator
It is untidy to have a fraction which has a surd denominator.
This can be 'tidied up' by multiplying the top and bottom of the fraction by a
particular expression. This is known as rationalising the denominator, since
surds are irrational numbers and so you are changing the denominator from an
irrational to a rational number.
Example
Rationalise the denominator of: a) 1 Ö2 .
b) 1 + 2 1  Ö2
a) Multiply the top and bottom of
the fraction by Ö2. The top will become
Ö2 and the bottom will become 2 (Ö2 times Ö2 = 2).
b) In situations like this,
look at the bottom of the fraction (the denominator) and change the sign (in
this case change the plus into minus). Now multiply the top and bottom of the
fraction by this.
Therefore:
1 + 2 
= 
(1 + 2)(1 + Ö2) 
= 
1 + Ö2 + 2 + 2Ö2 
= 
3 + 3Ö2 
1  Ö2 

(1  Ö2)(1 + Ö2) 

1 + Ö2  Ö2  2 

 1 
= 3(1 + Ö2)
Copyright © Matthew Pinkney 2003
