##
**Surds**

**Surd**is a number that does not have the exact value of square root of a rational number. All irrational numbers are

**surds**. For example: √3, √5, √(2/7) etc. are

**surds**.

###
**Types
of surds**

**: A surd which has only one irrational number such as √3, √13, etc is called pure surd.**

__Pure surds__**:The surd which has the product of rational and irrational number such as 5√2, 2√3, etc is called a mixed surd.**

__Mixed surd (compound surd)__**: Two or more surds of the same order are called like surds such as 2√3, 3√5.**

__Like surd__**: Two or more surds, which have different order are called unlike surds. For example,**

__Unlike surd__###
**Rationalization
of surd**

Surd can be rationalized by a certain surd to remove the root
sign in surd. Look at the following examples:

a. √2 is a surd. When we multiply by √2, then
the product is in rational form. i.e. √2×√2 = 2 which is a rational number.

b. When (√3 - √2) is multiplied by √3 + √2
then the product is in ratinal form. i.e. (√3 - √2) (√3 + √2) = 3 – 2 = 1 which
is rational number. So, (√3+√2) is rationalization factor or conjugate factor
of (√3-√2).

###
**Rules
of surds**

**1.**:

__Addition and subtraction__
Like surds can be added or subtracted. For example:

i. 5√3 + 2√3 = 7√3

ii. 7√2 - 3√2 = 4√2

**2.**:

__Multiplication and division__
When two or more surds have the same order, we multiply
or divide the internal value of surds. For example:

i. √5 × √2 = √(5×2) = √10

ii. √10 ÷ √2 = √(10÷2) = √5

###
**Properties
of surds**

i. The sum or difference of a rational and an
irrational number is a irrational number. For example: (3 + √5) and (4 - √2) are irrational numbers.

ii. The product of a rational and irrational
number is always an irrational number. It is also called a mixed surd. For example:
3×√2 = 3√2 is an irrational number.

iii. If a+√b = c+√d, where a and c are rational
numbers and √b and √d are irrational numbers, then a = c and b = d.

iv. If a+m√b = c+n√b, where a, c, m, n are
rational numbers and √b is an irraitonal number, then a = c and m = n.

###
*Workout
Examples*

*Workout Examples*

*Example 1: Identify the following pure or mixed surds.*

*Example 2: Write the rationalizing factor of:*

*a.*

*√8*

*b.*

*√3+√2*

*c.*

*5√3 - 1*

*Solution:*

*a. √8*

*= √2×2×2*

*= 2√2*

*∴*

*Rationalizing factor = √2*

*b. √3 +√2*

*∴*

*Rationalizing factor = √3 - √2*

*c. 5√3 - 1*

*∴*

*Rationalizing factor = 5√3 + 1*

*Example 3: Rationalize the denominator of the following:*

*Example 4: Simplify the following**You can comment your questions or problems regarding the surds here.*

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