Rules of Surd – Basic Forms of Surd, Similar Surds, Conjugate Surds

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.

1. √(a X b ) = √a X √ b
2. √(a / b ) = √a  / √b
3. √(a + b ) ≠ √a + √b
4. √(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.

1. 3√n and 5√ n
2. 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