Surds

Surds are numbers left in 'square root form' (or 'cube root form' etc).

Addition and subtraction of surds

aÖb + cÖb = (a + c)Öb
aÖb - cÖb = (a - c)Öb

Examples

4Ö7 - 2Ö7 = 2Ö7.
5Ö2 + 8Ö2 = 13Ö2

NB1: 5Ö2 + 3Ö3 cannot be manipulated because the surds are different (one is Ö2 and one is Ö3).
NB2: Öa + Öb is not the same as Ö(a + b) .

Multiplication and Division

Öab = Öa × Öb
Ö(a/b) = Öa
             Öb

Examples

Ö5 × Ö15 = Ö75
= 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 surd. 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). Doing this forms the conjugate of the denominator. 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)