Calculating Exactly with Surds

# Calculating Exactly with Surds

GCSE(H),

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 using a number of rules:

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 are 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: sqrt(frac(72)(27)) = sqrt(frac(36 times 2)(9 times 3)) = frac(6 times sqrt(2))(3 times sqrt(3)) = 2frac(sqrt(2))(sqrt(3)) = frac(2)(3)sqrt(2)sqrt(3)

## Examples

1. Simplify √300 + √27.

√300 + √27

= √100 x √3 + √9 x √3

= 10√3 + 3√3

= 13√3

2. Expand √200(1 + √2).

Answer: 10√2 + 20

√200(1 + √2)

= √200 + √200√2

= √100√2 + √400

= 10√2 + 20