Table of contents Surds Laws Of Surds. Comparison of surds Addition and Subtraction of Surds Multiplication of Surds Division of Surds Rationalising Factor Conjugate Surd

## 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  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) 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

## Laws Of Surds.

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

### Order And Base Of Surds

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

is a cubic 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  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.

### Conversion of mixed surds into pure surds.

Example: Express each of the following as pure surds

Solution:

### 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

Solution

### 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

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 ………. xn

= 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

n= 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

## 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  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

### 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

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

## Addition and Subtraction of Surds

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:

## Multiplication of Surds

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

## Division of Surds

a. Surds of the same order may  be divided

e.g 4

## 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.

## Conjugate Surd

are conjugate surds

The document Surd | Quantitative Aptitude for SSC CGL is a part of the SSC CGL Course Quantitative Aptitude for SSC CGL.
All you need of SSC CGL at this link: SSC CGL

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## FAQs on Surd - Quantitative Aptitude for SSC CGL

 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.

## Quantitative Aptitude for SSC CGL

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