What are Surds ? If x is a positive rational number and n is a positive integer such that x
1/n , n√x is irrational; then x
1/n is called a surd or radical. Surds are irrational numbers written in the form of roots of rational number.
The symbol √ ,
n√ is called the radical sign. Radicand is the expression written beneath the radical sign.
e.g. √8, here 8 is called the radicand.
Examples :- i) √2 ,
3√5,
7√9 are surds because 2, 5 and 9 are positive rational numbers and √2 ,
3√5,
7√9 are irrational numbers.
ii) But √4 ,
3√27 ,
5√32 are not surds as √4 = 2,
3√27= 3 ,
5√32 = 2 are rational numbers.
Every surds is an irrational number, but every irrational number is not surd. For example ¶ is an irrational number but not a surd
Is √225 x √4 is a surd or not ?
Solution :- √225 x √4 = √(15 x 15) x √(2 x 2) = 15 x 2 = 30 which is a rational number.
Therefore √225 x √4 is not a surd.
Addition and Subtraction of Surds Simplify and Write the following expressions in simplest form.
7√3 + 4√27 - √12
= 7√3 + 4√(9 x3) - √(4 x 3)
= 7√3 + 12√3 - 2√3
= (7 + 12 - 2) √3
= 17√3
Multiplication and Division of Surds Examples
(7√3 - 4) (5√3 +1)
= 7√3 (5√3 +1) - 4 (5√3 +1)
= 35 x 3 + 7√3 - 20√3 - 4
= 105 - 13 √3 - 4
= 101 - 13√3
Rationalisation of Surds When a surd is multiplied by another surd to obtain a rational number, it is called rationalisation. Each surd is called the rationalising factor of the other.
Rationalise the denominator
Example
4 + √3 2 + √3
= (
4 + √3) ( 2 - √3 ) (2 + √3) ( 2 - √3 )
=
4 ( 2 - √3 ) + √3 ( 2 - √3 ) (2)
2 - (√3)
2 =
8 - 4√3 + 2 √3 - 3 4 - 3
= 5 - 2 √3