# The Laws of Indices

## Introduction to Indices

An index or power is a small number placed above and to the right of a base number. The index indicates how many times the base number should be multiplied by itself.

For example, can be written as . The base is , and the exponent is . It is read as “ to the power of .”

Let’s look at some more examples:

• • • ## Laws of Indices

### Multiplying indices

When multiplying indices with the same base, we add the powers: Examples:

• • • ### Dividing indices

When dividing indices with the same base, we subtract the powers: Examples:

• • • • ### Power of a power

When raising an index to another power, we multiply the powers: Examples:

• • • • • ### Power of a product

When raising a product to a power, we distribute the power to each factor: Examples:  ### Power of zero

Any non-zero number raised to the power of is equal to : (where )

Examples:

• • • ### Negative indices

A negative index represents the reciprocal of the positive index: Examples:

• • ## Using the Laws of Indices to Simplify Numbers and Equations

We can use the laws of indices to simplify numbers and calculations. For example:

If , then    Let’s look at two more examples:

### Example

Simplify:

1. 2. 3. 4. 5. 6. 1. 2. 3. 4. 5. 6. ### Example

Simplify     