Type of surd or irrational root definition and example

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Type of Surd ( irrational root) definition and example:

Surd mean in mathematic irrational root, irrational root term, condition, type detail one by one detail

(1)Monomial surd or monomial irrational root

A “Surd” which contain a single term is called monomial surd.

Example:

√ 2,    √3,    √ a,    √ x     etc.

(2)Binomial Surd or binomial irrational root

A surd which contain sum of two monomial surd or sum of a

monomial surd and rational number is called a binomial surd.

Example:

√ 3 + √ 7   or √ 2 + 5   or  √ 11  – 8   etc.

(3) Product of two surd

if the product of two surd is a rational number, then each surd is called the rationalizing factor of the other.

(4) Rationalization of the given surd:

The process of multiplying a given surd by its rationalizing factor to get a rational number as product is called rationalization of the given surd.

(5) conjugate surd:

Two binomial surd of second order differing only in sign connecting their term are called conjugate surd. Thus, (√ a + √ b)  and

(√ a – √ b)  are conjugate of each other.

Example:

The conjugate of x + √ y     is    x   –  √  y.

The product of the conjugate of the surd  (√ a + √ b)  and

(√ a – √ b).

(√ a + √ b) (√ a – √ b) = (√ a) ² – (√ b) ²    =   a  –  b

is a rational quantity independent of any radical.

Example:2

The product of   a + b √ m   and its conjugate a – b √ m has no radical.

By numeric number

(3  + √ 5) (3 – √  5)   = (3) ²  –  (√ 5) ²  =  9  –  5 =   4

Rationalizing a Denominator:

Keeping the above discussion in mind, we observe that, in order to rationalize

a denominator of the form a  + b √  x   or    a   –  b √  x. We multiply both nominator and denominator by the conjugate factor a  – b √  x   or    a   +  b √  x. By doing this we eliminate the radical and thus obtained  a denominator free of any surd.

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