Surds are irrational numbers. They are the root of rational numbers whose value can not be expressed as exact fractions. Examples of surds are: √2, √7, √12, √18, etc.

- √(a X b ) = √a X √ b
- √(a / b ) = √a / √b
- √(a + b ) ≠ √a + √b
- √(a – b ) ≠ √a – √b

**Basic Forms of Surds**

√a is said to be in its basic form if A does not have a factor that is a perfect square. E.g. √6, √5, √3, √2 etc. √18 is not in its basic form because it can be broken into √ (9×2) = 3√2. Hence 3√2 is now in its basic form.

**Similar Surds**

Surds are similar if their irrational part contains the same numerals e.g.

- 3√n and 5√ n
- 6√2 and 7√2

**Conjugate Surds **

Conjugate surds are two surds whose product result is a rational number.

(i)The conjugate of √3 – √5 is √3 + √5

The conjugate of -2√7 + √3 is -2√7 – √3

In general, the conjugate of √x + √y is √x – √y

The conjugate of √x – √y = √x + √y

**Simplification of Surds **

Surds can be simplified either in the basic form or as a single surd.

Examples 1. Simplify the following in its basic form (a) √45 (b) √98

Solution

(a) √45 = √ (9 x 5) = √9 x √5 = 3√5

(b) √98 = √ (49 x 2) = √49 x √2 = 7√2

Example 2. Simplify the following as a single surd (a) 2√5 (b) 17√2

Solution

(a) 2√5 = √4 x √5 = √ (4 x 5) = √20

(b) 17√2 = √289 x √2 = √ (289 x 2) = √578