Math՚s: Exponential Notation: Comparison of Surds and Simplest Form (For CBSE, ICSE, IAS, NET, NRA 2022)

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Comparison of Surds

To know which surd is greater and which one is smaller, firstly we make them of equal orders and then compare their radicands together with their coefficients.

For Example- Compare and

Simplest Form of a Surd

A surd is said to be in its simplest form if it has-

  • smallest possible order
  • no fraction under radical (root) sign
  • no factor of the form , where is a positive integer, under the radical sign of order

For Ex- in simplest form will be

Basic Mathematical Operations on Surds

Addition and Subtraction- To add or subtract surds, they must be similar or like surds. If they are not, they are first made similar by making their radicands and order same. When their radicands and order are same, their coefficients are added or subtracted.

Example – (i)


Multiplication and Division- Two surds can be multiplied or divided if they are of the same order. So, before multiplying or dividing, we change them to the surds of the same order.

Example- (i)


Rationalization of Surds

Conversion of surds into rational numbers is known as rationalization of surds. For converting a surd into a rational number, we need to multiply it with another surd. In such a case, each surd is called the rationalizing factor of the

another surd.

For Example- (i) Find the rationalizing factors of and .

∴ Rationalizing factor is because on multiplying with , we get

which is a rational number.


∴ Rationalizing factor is

Rationalizing the denominator of a fraction- Rationalization is usually done of the denominator of an expression involving irrational surds.

Ex- (i) Rationalize the denominator of


(ii) Rationalize


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