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.
- √(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
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.
