Surds: Rules, Basic forms, simplification, Addition and Subtraction, Multiplication and Division, Rationalization and Equality of Surds

CONTENT

• Rules of surds
• Basic Form of Surds
• Similar Surds
• Conjugate Surds
• Simplification of Surds
• Additional & Subtraction of Surds
• Multiplication and Division of Surds
• Rationalization of Surds
• Equality of Surds

Rules of Surds

 Surds are irrational numbers. They are the root of rational numbers whose value cannot 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

 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

Examples

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

Addition and Subtraction of Surds

 Surds in their basic forms which are similar can be added or subtracted.

Examples

Evaluate the following

(a)√32 + 3√8       (b) 7√3 – √75      (c) 3√48 – √75 + 2√12

Solution

•  (√32  + 3√8

          = √ (16 x 2) + 3√ (4 x 2)

          =4√2 + 6√2

          = 10√2

     (b) 7√3 – √75

= 7√3 – √ (25 x 3)

=7√3 – 5√3     =2√2

     (c) 3√48 – √75 + 2√12

          = 3√ (16 x 3) – √ (25 x 3) + 2√ (4 x 3)

          = 12√3 – 5√3 + 4√3

          = 11√3

Evaluation

1. Simplify the following (a) 5√ 12 – 3√ 18 + 4√72 + 2√75     (b) 3√2 – √32 + √50 + √98

2. Simplify the following as a single surd (i) 8√3      (ii) 13√2

Multiplication and Division of Surds

Example: Evaluate the following (a) √45 x √28    (b) √24 /√50

Solution

(a) √45 x √28

= √ (9 x 5) x √ (4 x 7)

= 3√5 x 2√7

= 3 x 2 x √ (5 x 7)

= 6√35

(b)√24 / √50

          = √ (24 / 50)

          = √ (12 / 25)

          = √12 / √25

          = √ (4 x 3) / 5

          = 2√3 / 5

Evaluation:

 Simplify 1. √6 x (3 – √5)      2. (2√3 – √7)(2√3 + √7)

                2. Multiply the following by their conjugate (a) √3 – 2√5 (b) 3√2 + 2√3

Surds Rationalisation

Rationalisation of surds means multiplying the numerator and denominator by the denominator or by the conjugate of the denominator.