CLAT  >  Quantitative Techniques for CLAT  >  Surd

Surd - Quantitative Techniques for CLAT


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 | Quantitative Techniques for 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 | Quantitative Techniques for 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 | Quantitative Techniques for CLAT is an irrational number. So Surd | Quantitative Techniques for CLAT is a surd.

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

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

iii)Surd | Quantitative Techniques for CLAT is not a surd because  2 + Surd | Quantitative Techniques for 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 | Quantitative Techniques for CLAT 

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

Surd | Quantitative Techniques for CLAT  

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

Surd | Quantitative Techniques for CLAT 

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

Surd | Quantitative Techniques for CLAT

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

Surd | Quantitative Techniques for CLAT

 

Order And Base Of Surds

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

 Surd | Quantitative Techniques for CLAT  is a cubic surd

 Surd | Quantitative Techniques for 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 | Quantitative Techniques for 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 | Quantitative Techniques for CLATSurd | Quantitative Techniques for CLAT etc are mixed surds.

Conversion of mixed surds into pure surds. 

 Example: Express each of the following as pure surds

Surd | Quantitative Techniques for CLAT 

Solution:

Surd | Quantitative Techniques for CLAT 

Surd | Quantitative Techniques for CLAT 

Surd | Quantitative Techniques for CLAT

 Surd | Quantitative Techniques for CLAT

 Surd | Quantitative Techniques for 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 | Quantitative Techniques for CLAT

 Solution

Surd | Quantitative Techniques for CLAT
Surd | Quantitative Techniques for CLAT
Surd | Quantitative Techniques for 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 | Quantitative Techniques for CLAT

 

Example 1

Convert  Surd | Quantitative Techniques for 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 | Quantitative Techniques for CLAT

Example 2

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

Solution

Here n = 2

M = 8

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

So Surd | Quantitative Techniques for CLAT

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

Solution:

Here n = 3

M = 6

K = m/n = 6/3 = 2

So Surd | Quantitative Techniques for CLAT

 

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

Solution:

Surd | Quantitative Techniques for CLAT

 

Now n = 3

       m = 6

 k = Surd | Quantitative Techniques for CLAT

So given surd = Surd | Quantitative Techniques for CLAT     

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

Solution: 

Surd | Quantitative Techniques for CLAT

Surd | Quantitative Techniques for CLAT

Here n = 2

        M = 4

Surd | Quantitative Techniques for CLAT 

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

Step 1.Let the surds be Surd | Quantitative Techniques for 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 | Quantitative Techniques for CLAT  and

    Step IV Write the requested surd as

Surd | Quantitative Techniques for CLAT 

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

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

            n= order of Surd | Quantitative Techniques for CLAT = 6

 n = LCM of 4 and 6 – 12

Now m1 = Surd | Quantitative Techniques for CLAT 

So Surd | Quantitative Techniques for CLAT

 and  Surd | Quantitative Techniques for CLAT

So required surds are Surd | Quantitative Techniques for CLAT

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

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

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

   n = LCM of 2, 3, = 6.

Now m1 =
Surd | Quantitative Techniques for CLAT Hence 

    Thus reqd. surds are Surd | Quantitative Techniques for 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 | Quantitative Techniques for CLAT or Surd | Quantitative Techniques for CLAT

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

   Since 35 > 26 so Surd | Quantitative Techniques for CLAT

 

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

Solution: Surds are of the same order

   So Surd | Quantitative Techniques for 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 | Quantitative Techniques for CLAT

Solution: Order of the two surds is 3, 4

   Their LCM is 12

Surd | Quantitative Techniques for CLAT

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

n1 = 2   n2 = 3

n = LCM of 2, 3 = 6

Surd | Quantitative Techniques for CLAT
Surd | Quantitative Techniques for CLAT

 

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

Solution: Given surd are Surd | Quantitative Techniques for CLAT

   The order of these surds is 2, 3, 6

  LCM of 2, 3, 6 is 6

 The surds can be written as

   Surd | Quantitative Techniques for CLAT

Surd | Quantitative Techniques for CLAT

Surd | Quantitative Techniques for CLAT

Surd | Quantitative Techniques for CLATas it is

Now compare the radiants

Surd | Quantitative Techniques for CLAT

Addition and Subtraction of Surds

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

Solution: Surd | Quantitative Techniques for CLAT

Surd | Quantitative Techniques for CLAT 

Example 2: Simplify Surd | Quantitative Techniques for CLAT

Solution:

Surd | Quantitative Techniques for CLAT

Surd | Quantitative Techniques for CLAT

Surd | Quantitative Techniques for CLAT

Surd | Quantitative Techniques for CLAT

Example 3: Simplify 2 Surd | Quantitative Techniques for CLAT

Solution: 

Surd | Quantitative Techniques for CLAT
Surd | Quantitative Techniques for CLAT

 

Multiplication of Surds

Surds of the same order can e multiplied

Surd | Quantitative Techniques for CLAT

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

Surd | Quantitative Techniques for CLAT

Division of Surds

a. Surds of the same order may  be divided

            Surd | Quantitative Techniques for CLAT

e.g 4

Surd | Quantitative Techniques for CLAT

Surd | Quantitative Techniques for 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 | Quantitative Techniques for CLAT

Surd | Quantitative Techniques for CLAT

Conjugate Surd

Surd | Quantitative Techniques for CLAT are conjugate surds

The document Surd | Quantitative Techniques for CLAT is a part of the CLAT Course Quantitative Techniques for CLAT.
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FAQs on Surd - Quantitative Techniques for CLAT

1. What are surds?
Ans. Surds are mathematical expressions that involve irrational numbers or roots. They are typically represented in the form of square roots, cube roots, or higher roots that cannot be simplified into rational numbers.
2. What are the laws of surds?
Ans. The laws of surds include the following: - The product of two surds with the same root is equal to the surd of the product of their radicands. - The division of two surds with the same root is equal to the surd of the division of their radicands. - The sum or difference of two surds with the same root is obtained by adding or subtracting their radicands.
3. How do you compare surds?
Ans. To compare surds, you need to compare their radicands. If the radicand of one surd is greater than the radicand of another surd, then the first surd is greater. If the radicands are equal, then the surds are equal.
4. How do you perform addition and subtraction of surds?
Ans. To add or subtract surds, you need to have the same root in both surds. If the roots are the same, you can simply add or subtract the radicands. If the roots are different, you need to rationalize the surds by multiplying both the numerator and denominator by the conjugate of the denominator.
5. How do you multiply and divide surds?
Ans. To multiply surds, you can simply multiply the radicands together. To divide surds, you can divide the radicand of the numerator by the radicand of the denominator. However, if the denominator is a surd, you need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
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