# Indices & the Law of Indices

## Introduction

Indices are a useful way of more simply expressing large numbers. They also present us with many useful properties for manipulating them using what are called the

**Law of Indices**.## What are Indices?

The expression 2

^{5}is defined as follows:
We call "2" the

**base**and "5" the**index**.## Law of Indices

To manipulate expressions, we can consider using the Law of Indices. These laws only apply to expressions with the

**same base**, for example, 3^{4}and 3^{2}can be manipulated using the Law of Indices, but we cannot use the Law of Indices to manipulate the expressions 3^{5}and 5^{7}as their base differs (their bases are 3 and 5, respectively).## Six rules of the Law of Indices

**Rule 1:**

Any number, except 0, whose index is 0 is always equal to 1, regardless of the value of the base.

**An Example:**

Simplify 2

^{0}:Rule 2: |

**An Example:**

Simplify 2

^{-2}:**Rule 3:**

To multiply expressions with the same base, copy the base and add the indices.

**An Example:**

Simplify :

*(note:*5 = 5^{1}*)***Rule 4:**

To divide expressions with the same base, copy the base and subtract the indices.

**An Example:**

Simplify | : |

**Rule 5:**

To raise an expression to the nth index, copy the base and multiply the indices.

**An Example:**

Simplify (y

^{2})^{6}:Rule 6: |

**An Example:**

Simplify 125

^{2/3}: