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)`

Simplify √300 + √27.

Using a prime factor tree, 300 = 2^{2} x 5^{2} X 3

Therefore `sqrt(300) = sqrt(2^2 xx 5^2 xx 3)`

`= sqrt(2^2) xx sqrt(5^2) xx sqrt(3)`

`= 2 xx 5 xx sqrt(3)`

`= 10sqrt(3)`

And `sqrt(27)`

`= sqrt(9 xx 3)`

`= sqrt(3^2 xx 3)`

`= sqrt(3^2) xx 3`

`= 3sqrt(3)`

Adding both together:

`= 10√3 + 3√3`

`= 13√3`

Answer: 13√3

Expand √200(1 + √2).

√200(1 + √2)

= √200 + √200√2

= √100√2 + √400

= 10√2 + 20

Answer: 10√2 + 20

See also Working with Roots in Algebra

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