# Math's: Exponential Notation: Surds, Their Types and Order and Laws of Surds

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## Surds

A surd is a positive irrational number of the type , where it is not possible to find exactly the nth root of , where is a positive rational number. For Ex-is a surd as we cannot precisely find out the cube root of 2.

In the surd , the symbol is called a radical sign. The index is called the order of the surd and is called the radicand. Thus, in the above example, 3 is the order 2 is the radicand.

## Types of Surds

A surd, with rational factor 1 only, another factor being irrational is called a pure surd.

For example, and are pure surds.

A surd, having rational factor other than 1 along with the irrational factor, is called a mixed surd.

For example, and are mixed surds.

Two surds that can be reduced to the same irrational factor are said to be similar or like surds.

For example- and are like surds.

## Laws of Surds

(i)

(ii)

(iii)

(iv)

(v)

(vi)

1. State which of the following is a surd and also state their type (pure or mixed) in case of surds.

(i) (ii) (iii)

Solution (i) is the square root of a perfect square.

is not a surd.

(ii) is the cube root of an imperfect cube. Therefore, it is a surd. is a mixed surd as it has one more rational factor 5 besides 1.

(iii) is an irrational number. So, it is a surd. has no rational factor other than one, so it is a pure surd.

2. Find the ‘order’, ‘radicand’ and ‘coefficient’ of each of the following surds.

(i) (ii) (iii)

Solution: (i) Order- 4, Radicand- 214, Coefficient- 1

(ii) Order- 2, Radicand- 41, Coefficient- 2 (When order is not given, it is assumed to be 2)

(iii) Order- 3, Radicand- 18, Coefficient- 7

3. Apply the laws of surds and simplify the following-

(i) (ii) (iii)

Solution (i)

(ii)

(iii)