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.
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 .
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