Calculating Exactly with Surds

## Calculating Exactly with Surds

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)

## Example 1

Simplify √300 + √27.

Using a prime factor tree, 300 = 22 x 52 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)

= 10√3 + 3√3

= 13√3

## Example 2

Expand √200(1 + √2).

√200(1 + √2)

= √200 + √200√2

= √100√2 + √400

= 10√2 + 20