Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

If a is a rational number and n is a positive integer such that the nth root of a i.e. a^1/n or  is an irrational number, then  a^1/n is called a surd. It is also called a radical of order n, a is called the radicand.

Example: 

i) is a surd. It can also be written as (3)^1/2, Since 3 is a rational number and 2 is a positive integer and  is an irrational number. So  is a surd.

 ii) is not a surd

Because 8 is a rational number and 3 is a +ve integer but   =2 is not an irrational number. So  is not a surd

iii) is not a surd because  2 +  is not a rational number.

 iv) Every surd is an irrational number but every irrational number is not a surd

i) For every positive integer n and a + ve rational number a  

ii) If n is a positive integer and a, b are rational numbers then

 iii) If n is a + ve integer and a, b are rational numbers then

iv) If m and n are positive integers and a is a positive rational number then

 v) If m and n are positive integers and a is a positive rational number then

Surd  is a surd of nth order with base a.  is a quadratic surd

   is a cubic surd

  is a biquadratic surd

A surd which does not have a rational factor other than unity (1) is called a pure surd. For example  etc. are pure surds.

A surd which has a rational factor other than unity, the other factor being rational is called a mixed surd e.g.  etc are mixed surds.

 Example: Express each of the following as pure surds

Solution:

 Example 1: Express each of the following as mixed surds in its simplest form

 Solution

 step I Obtain the order of given surd say n

step II Obtain the order of new surd into which the given surd is to be converted say m

step III compute m/n let m/n = k

step IV write the new surd as

Example 1

Convert   into a surd of order 6

Solution: n = order of given surd = 2

m= order of new surd = 6

Let K = m/n = 6/2 = 3

So

Example 2

Convert  into a surd of order 8

Solution

Here n = 2

M = 8

K = m/n = 8/2 = 4.

So

Example 3: Convert  into a surd of order 6

Solution:

Here n = 3

M = 6

K = m/n = 6/3 = 2

So

Example 4: Express 2 x   as pure Surd of order 6

Solution:

Now n = 3

       m = 6

 k =

So given surd =      

 Example 5: Express  as pure surd of order 4

Solution: 

Here n = 2

        M = 4

Conversion of two or more surds into surds of the same order.

Step 1.Let the surds be

So that then order are n1, n2, n3 …….. etc

Step II Find out LCM of n₁, n₂, n₃ ………. x_n

                                    = n (say)

Step III Compute   and

    Step IV Write the requested surd as

Example 1: Convert  into surds of the same but smallest order

Solution: n₁ = order of  =4

            n₂ = order of  = 6

 n = LCM of 4 and 6 – 12

Now m₁ =  

So

 and  

So required surds are

 Example 2: Convert  and into surds of same but smallest order

n₁ = order of the surd

n₂ = order of the surd

   n = LCM of 2, 3, = 6.

Now m₁ =
 Hence 

    Thus reqd. surds are

If the surds are of the same order, they can be compared by comparing their radicands.

 Example 3: Which surd is larger  or

Solution: Given surds are of the same order and their radicands are 26 and 35 respectively

   Since 35 > 26 so

Example 4: Arrange the following surds in ascending order

Solution: Surds are of the same order

   So

We first reduce them to the same but smallest order and then compare their radiants.

 Example 5 : Which surd is larger

Solution: Order of the two surds is 3, 4

   Their LCM is 12

Example 6: Which is greater

n₁ = 2   n₂ = 3

n = LCM of 2, 3 = 6

Example 7: Arrange the following surds in ascending order of magnitude

Solution: Given surd are

   The order of these surds is 2, 3, 6

  LCM of 2, 3, 6 is 6

 The surds can be written as

as it is

Now compare the radiants

 Surds having same irrational factor are called similar surds. e.g.  are similar surds. such type of surds can be added or subtracted. Where as unlike surds having no common irrational factor cannot be added or subtracted.  

 If the surds are unlike, reduce each of them to its simplest form and express each in such a way that they have a common irrational factor.

Example 1: Simplify

Solution:

Example 2: Simplify

Solution:

Example 3: Simplify 2

Solution: 

Surds of the same order can e multiplied

If the surds are not of the same order, they can be multiplied after converting them to surds of the same order

a. Surds of the same order may  be divided

e.g 4

 i) If the product of two surds is a rational number then each one of them is called the rationalizing factor of the other.

 are conjugate surds