# Indices & Surds: Definition of Indices

An **index** (plural: indices) is the power or exponent of a number.

For Example: has an index of 5.

Here, by the term we basically mean p x p x p x p x p i.e product of p 5 times.

Similarly, by the term we basically mean a x a x a … n times. i.e. product of a n times.

Here, a is called the **base **and n is called the** power or index** of the number .

In the term , a is the **base** and is the **power or index** of the number . It may also be called as **nth root **of a, denoted by .

The manipulation of indices can be a powerful tool to simplify an expression. There are various laws of indices which govern the whole operation.**1. Multiplication Rule: **

x = **For Example:**

x = =

Here, L.H.S = x = 27 x 81 = 2187

R.H.S = = 2187

L.H.S = R.H.S = 2187**2. Multiplication Rule: **

x x x …… = **For Example: **

x x = =

Here, L.H.S. = x x = 9 x 27 x 81 = 19683

R.H.S. = = 19683

L.H.S = R.H.S = 19683

**3. Division Rule: **

= **For Example: **

= = = 3

Here, L.H.S. = = =3

R.H.S. = = = 3

L.H.S. = R.H.S. = 3**4. Power Rule: **

= **For Example: **

=

Here, L.H.S. = = = 729

R.H.S. = = = 729

L.H.S. = R.H.S. = 729

**5. Power Rule: **

= **For Example: **

= = **6. Power Rule: **

= **For Example: **

=

7. = **For Example: **

=

8. = 1 (where x 0)**For Example: **

= 1

9. = x **For Example: **

= x = 9 x 25 = 225

Here, L.H.S. = = = 225

R.H.S. = x = 9 x 25 = 225.

10. = **For Example: **

= =

Here, L.H.S. = = = 0.5625

R.H.S. = = = 0.5625

L.H.S. = R.H.S. = 0.5625

11. If = and x -1, 0 , 1 then m = n.**For Example: **

=

Here, 3 -1, 0 , 1

So, m = n