CLAT  >  Quantitative Techniques for CLAT  >  Surd

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

1 Crore+ students have signed up on EduRev. Have you?


Surds

If a is a rational number and n is a positive integer such that the nth root of a i.e. a1/n or Surd - Notes | Study Quantitative Techniques for CLAT - CLAT is an irrational number, then  a1/n is called a surd. It is also called a radical of order n, a is called the radicand.

Example: 

i)Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 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 Surd - Notes | Study Quantitative Techniques for CLAT - CLAT is an irrational number. So Surd - Notes | Study Quantitative Techniques for CLAT - CLAT is a surd.

 ii)Surd - Notes | Study Quantitative Techniques for CLAT - CLAT is not a surd

Because 8 is a rational number and 3 is a +ve integer but Surd - Notes | Study Quantitative Techniques for CLAT - CLAT  =2 is not an irrational number. So Surd - Notes | Study Quantitative Techniques for CLAT - CLAT is not a surd

iii)Surd - Notes | Study Quantitative Techniques for CLAT - CLAT is not a surd because  2 + Surd - Notes | Study Quantitative Techniques for CLAT - CLAT is not a rational number.

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

 

Laws Of Surds.

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

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 

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

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT  

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

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 

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

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

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

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 

Order And Base Of Surds

Surd Surd - Notes | Study Quantitative Techniques for CLAT - CLAT is a surd of nth order with base a. Surd - Notes | Study Quantitative Techniques for CLAT - CLAT is a quadratic surd

 Surd - Notes | Study Quantitative Techniques for CLAT - CLAT  is a cubic surd

 Surd - Notes | Study Quantitative Techniques for CLAT - CLAT is a biquadratic surd

Pure surds and mixed surds.

A surd which does not have a rational factor other than unity (1) is called a pure surd. For example Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 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. Surd - Notes | Study Quantitative Techniques for CLAT - CLATSurd - Notes | Study Quantitative Techniques for CLAT - CLAT etc are mixed surds.

Conversion of mixed surds into pure surds. 

 Example: Express each of the following as pure surds

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 

Solution:

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

                

To express the given surds as mixed surds in its simplest form 

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

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 Solution

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT
Surd - Notes | Study Quantitative Techniques for CLAT - CLAT
Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Conversion of a surd into a surd of given order

 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 Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 

Example 1

Convert  Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 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 Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Example 2

Convert Surd - Notes | Study Quantitative Techniques for CLAT - CLAT into a surd of order 8

Solution

Here n = 2

M = 8

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

So Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Example 3: Convert Surd - Notes | Study Quantitative Techniques for CLAT - CLAT into a surd of order 6

Solution:

Here n = 3

M = 6

K = m/n = 6/3 = 2

So Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 

Example 4: Express 2 x Surd - Notes | Study Quantitative Techniques for CLAT - CLAT  as pure Surd of order 6

Solution:

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 

Now n = 3

       m = 6

 k = Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

So given surd = Surd - Notes | Study Quantitative Techniques for CLAT - CLAT     

 Example 5: Express Surd - Notes | Study Quantitative Techniques for CLAT - CLAT as pure surd of order 4

Solution: 

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Here n = 2

        M = 4

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 

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

Step 1.Let the surds be Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

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

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

                                    = n (say)

Step III Compute Surd - Notes | Study Quantitative Techniques for CLAT - CLAT  and

    Step IV Write the requested surd as

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 

Example 1: Convert Surd - Notes | Study Quantitative Techniques for CLAT - CLAT into surds of the same but smallest order

Solution: n1 = order of Surd - Notes | Study Quantitative Techniques for CLAT - CLAT =4

            n= order of Surd - Notes | Study Quantitative Techniques for CLAT - CLAT = 6

 n = LCM of 4 and 6 – 12

Now m1 = Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 

So Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 and  Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

So required surds are Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 Example 2: Convert Surd - Notes | Study Quantitative Techniques for CLAT - CLAT andSurd - Notes | Study Quantitative Techniques for CLAT - CLAT into surds of same but smallest order

n1 = order of the surd Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

n2 = order of the surd Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

   n = LCM of 2, 3, = 6.

Now m1 =
Surd - Notes | Study Quantitative Techniques for CLAT - CLAT Hence 

    Thus reqd. surds are Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Comparison of surds

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

 Example 3: Which surd is larger Surd - Notes | Study Quantitative Techniques for CLAT - CLAT or Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

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

   Since 35 > 26 so Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 

Example 4: Arrange the following surds in ascending order Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Solution: Surds are of the same order

   So Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 

Comparing the surds of distinct order

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

 Example 5 : Which surd is larger Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Solution: Order of the two surds is 3, 4

   Their LCM is 12

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Example 6: Which is greater Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

n1 = 2   n2 = 3

n = LCM of 2, 3 = 6

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT
Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 

Example 7: Arrange the following surds in ascending order of magnitude Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Solution: Given surd are Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

   The order of these surds is 2, 3, 6

  LCM of 2, 3, 6 is 6

 The surds can be written as

   Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Surd - Notes | Study Quantitative Techniques for CLAT - CLATas it is

Now compare the radiants

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Addition and Subtraction of Surds

 Surds having same irrational factor are called similar surds. e.g. Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 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.  

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 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 Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Solution: Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT 

Example 2: Simplify Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Solution:

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Example 3: Simplify 2 Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Solution: 

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT
Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

 

Multiplication of Surds

Surds of the same order can e multiplied

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

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

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Division of Surds

a. Surds of the same order may  be divided

            Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

e.g 4

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Rationalising Factor

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

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

Conjugate Surd

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT are conjugate surds

The document Surd - Notes | Study Quantitative Techniques for CLAT - CLAT is a part of the CLAT Course Quantitative Techniques for CLAT.
All you need of CLAT at this link: CLAT
56 videos|35 docs|91 tests
Use Code STAYHOME200 and get INR 200 additional OFF
Use Coupon Code
56 videos|35 docs|91 tests
Download as PDF

Download free EduRev App

Track your progress, build streaks, highlight & save important lessons and more!

Related Searches

MCQs

,

mock tests for examination

,

Free

,

video lectures

,

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

,

Previous Year Questions with Solutions

,

Objective type Questions

,

past year papers

,

pdf

,

Sample Paper

,

Extra Questions

,

study material

,

shortcuts and tricks

,

practice quizzes

,

ppt

,

Viva Questions

,

Important questions

,

Exam

,

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

,

Surd - Notes | Study Quantitative Techniques for CLAT - CLAT

,

Summary

,

Semester Notes

;