# Surds & Indices

**Surds **: which are square root and it cannot be simplified into rational numbers. But **indices** can be simplified.

For Example: (3 / 9)^{(1/2)} can be written
as 1/ but cannot be written in the form of fraction.

Irrational numbers which contain the radical sign ( ) are called as surds.

*Note :
*

1. All surds are irrational numbers.

2. All irrational numbers are not surds.

**Indices:** Indices
refers to the power to which a number is raised. Index is used to show that a
number is repeatedly multiplied by itself.

For example: a^{3 }is a number with
an index of 3 and base ‘a’. It is called as **“a to the power of 3” **

**IMPORTANT FACTS AND FORMULAE**

**1. Laws of Indices:**

**2. Surds:**

Let a be rational number and n be a positive integer such that

Then, is called a surd of order n.

**3. Laws of Surds:**

**4. Expressing a number in radical form**

Example: l x^{(m/n)} l =

The exponential form l x^{(m/n)} l is expressed in radical form as

**Important points to Remember**

1) Any number raised to the power zero is always equals to one. (Eg:
x ^{0} = 1)

2) Surd can be simplified if factor of x is a perfect square.

3) If denominator in a fraction has any surds, then rationalize the denominator by multiplying both numerator and denominator by a conjugate surd.

4) Every surd is an irrational number, but every irrational number is not a surd.

5) The conjugate of (2 + 7i) is (2 – 7i)

6) Different expressions can be simplified by rationalizing the denominator and eliminating the surd.

**Rationalizing the denominator:**

To rationalize the denominator multiply with its conjugate to both numerator and denominator

Example 1:

Example 2: