**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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