Powers

Introduction

A power, or an exponent, is used to write a product of numbers very compactly. In this leaflet we remind you of how this is done, and state a number of rules, or laws, which can be used to simplify expressions involving indices.

1. Powers, or exponents

We write the expression 3×3×3×3 as

We read this as "three to the power four". Similarly z × z × z =

We read this as " z to the power three" or " z cube".

In the expression the exponent is c and the number b is called the base. Your calculator will probably have a button to evaluate powers of numbers. It may be marked . Check this, and then use your calculator to verify that

Exercises

1. Without using a calculator work out the value of

2. Write the following expressions more concisely by using powers.

Answers

2. The laws of exponents

To manipulate expressions involving exponents we use rules known as the laws of exponents. The laws should be used precisely as they are stated - do not be tempted to make up variations of your own! The three most important laws are given here:

First law

When expressions with the same base are multiplied, the exponents are added.

Example

We can write

You could verify this by evaluating both sides separately.

Example

Second Law

When expressions with the same base are divided, the exponents are subtracted.

Example

We can write

Third law

Note that m and n have been multiplied to yield the new exponent mn.

Example

It will also be useful to note the following important results:

Exercises

1. In each case choose an appropriate law to simplify the expression:

2. Use one of the laws to simplify, if possible,

Answers

2. This cannot be simplified because the bases are not the same.