Basic surd manipulation
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. Leaving them as surds is more mathematically precise.

Addition and subtraction of surds:
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 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). 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)