SURDS AND RADICALS math capsule

Today we are going to learn about surds and radicals  in this blog post .

Before beginning with 'surds and radicals' you should visit and revise the concepts of number system ,

NUMBER SYSTEM

So, lets begin with surds and radicals

 Let x be a rational number and n be a positive number such that ⁿ√x is  irrational , then ⁿ√x is called a radical of order n or surd and here x is called the radicand.

* A surd of order 2 is called quadratic surd, i.e. √2
* A surd of order 3 is called a cubic surd i.e. ³√5
* A surd of order 4 is called a biquadratic surd
* A surd has no fraction under the radical sign .
* The radicand has no factor with exponent n.
* A surd is not equal to any surd of order lower than n.
* ⁿ√x is a surd only if x is a rational and ⁿ√x is irrational.
* When x is irrational or ⁿ√x is rational then ⁿ√x is not a surd.



Laws of radicals -

The laws of indices which are applicable to the surds also are,

(a)  (ⁿ√x)ⁿ = x

(b) ⁿ√xy = ⁿ√x ⁿ√y

(c) ⁿ√(x / y ) = ⁿ√x / ⁿ√y

(d) (ⁿ√x)ⱽ = ⁿ√xⱽ

(e) ⱽ√ⁿ√x = ⱽⁿ√x = ⁿ√ⱽ√x


TYPES OF SURDS

1. PURE SURDS
A surd which only 1 as a rational factor the other factor being irrational is called a pure surd. e.g. √2

2. MIXED SURDS
A surd which is not a pure surd or has factor other than unity, the other factor being irrational is called a mixed surd. e.g. 2√3

3. SIMILAR (LIKE) SURDS
If two or more surds having the same irrational factor are known as similar / like surds .
e.g. , √3 , 8√3

4. DISSIMILAR SURDS 
If two or more surds have different irrational factor are known as dissimilar surds.
e.g.,√3 , √2 are unlike surds.


OPERATIONS ON SURDS 

ADDITION OF LIKE SURDS

Like surds are added by using laws of numbers.
e.g.

√2 + 3√2 = 4 √2 

SUBTRACTION OF LIKE SURDS

Like surds are subtracted by using laws of numbers.
e.g.

5√x - 2√x = 3√x 

MULTIPLICATION AND DIVISION OF SURDS

ⁿ√x *  ⁿ√y = ⁿ√xy

ⁿ√x / ⁿ√y= ⁿ√(x / y)

RATIONALISATION OF SURDS 

Process of converting surd into rational number is called as rationalisation of the surds.
When the product of two surds is a rational number, then each surd is called a rationalising factor of the other .

 ⁿ√x * ⁿ√y = ⁿ√x*y = ⁿ√xⁿ = x

MONOMIAL OF SURDS

A surd having only one term is known as a monomial surd e.g √2 

BINOMIAL OF SURDS

A surd consisting of the sum of two monomial surds or the sum of a monomial surd and a rational number ,
e.g.  (√5 + √3 )

TRINOMIAL SURDS

A surd consisting of the sum of three monomial surds or two binomials.
e.g.  (√5 + √3 + √2 )

CONJUGATE SURDS 

The binomial surds which differ only in the sign between them are called as conjugate surds.
e.g (x + √y ) and ( x - √y ) are conjugate surds .


THANKS FOR READING THIS AT Math Capsule

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