A surd is an irrational number. It cannot be expressed either as an integer or as a fraction.

√2 is a surd. However, √4 is not a surd, as the square root of 4 is an integer (2).

Surds should be expressed in their lowest value. This involves manipulating a surd by adding, multiplying or dividing surds:

`sqrt(ab) = sqrt(a) xx sqrt(b)` (multiplying two surds);

`msqrt(a) + nsqrt(a) = (m + n)sqrt(a)` (adding two like surds);

`sqrt(frac(a)(b)) = frac(sqrt(a))(sqrt(b))` (dividing two surds).

Simplifying a surd involves identifying the factors of a surd, determining which of these factors are square numbers, then manipulating the surds using the rules above. Using a Prime Factor Tree can often quickly identify factors that are square numbers. Any square numbers can then be turned into their roots:

Multiplying two surds: `sqrt(72) = sqrt(36) xx sqrt(2) = 6sqrt(2)`

Adding two like surds: `3sqrt(3) + 5sqrt(3) = 8sqrt(3)`

Dividing two surds: `frac(sqrt(72)sqrt(2))(sqrt(24)) = frac(sqrt(144))(sqrt(24)) = sqrt(frac(144)(24)) = sqrt(6)`