Surds


What are Surds ?

If x is a positive rational number and n is a positive integer such that x1/n ,  n√x  is irrational; then  x1/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  
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