^{n}

Here ‘a’ is called the base and ‘n’ is known as the index of the power. Basically it is an exponential expression.

## Rules of Indices: -

Indices: - Any root of a non negative rational number, which can not be found or does not provide an exact solution. For example

^{n}√a

Here ‘a’ is called as radicand and ‘n’ is known as the order of surds. ‘n’ should be a natural number.

**Note: - All surds are irrational number but all irrational number are not surds.**

### Order of the surds:

**Note: -**

**a)**Surds of 2

^{nd}order are known as quadratic surds.

**b)**Surds of 3

^{rd}order are known as cubic surds.

## Rules of Surds: -

### Type of Surds: -

**Pure Surds:-**Those surds which do not have factor other than 1. For example^{2}√3,^{3}√7**Mixed Surds:-**Those surds which do not have factor than 1. For example**√27 = 3√3, √50 = 5√2****Similar Surds:-**When the radicands of two surds are same. For example**5√2 and 7√2****Unlike Surds:-**When the radicands are different. For example**√2**and**2√5**

### Formulas/ Patterns Including Short Tricks:

#### 1) Arrangement of surds in either increasing or decreasing order –

First of all, take the LCM of the denominator of the powers and use it to make them same and then solve it.**Example: -**

#### 2) Calculation on indices/ surds to find the value of any expression –

In such questions, factorize the expression in the smallest possible number then solve the expression using the rules of surds/ indices.**Example: -**

#### 3) Addition operation on surds –

In such questions, break the x into m (m + 1) form then answer would be (m + 1).

**Example: -**

#### 4) Subtraction operation on surds –

In such questions, factorize the x into m (m + 1) form then ‘m’ would be the answer.

**Example: -**

#### 5) Multiplication operation on surds –

In such questions, answer would be

Where n = number of time x is repeated.

**Example: -**

#### 6) Rationalization on surds –

In this process, we convert the denominator of the surds into a rational number. For this we have to multiply both numerator and denominator of the surds with another surd and obtain rational number by applying formula (a + b) * (a – b) = (a^{2}- b

^{2})

**Example: -**

Find the value of