# Power of Power Rule for Exponents

Image Source: http://cdn.abclocal.go.com

The world’s population is exponentially increasing, especially in the world’s developing Countries.

Population Growth follows the Power of Power Rule, as shown in the example below.

Image Copyright 2013 by Passy’s World of Mathematics

In this lesson we look at how to expand a single exponent item that is in brackets and raised to a Power.

There is a shortcut fast track rule for these expressions which involves multiplying the power values.

This rule is called the “Power of Power” Rule.

But first let’s look at expanding Power of Power without using this rule.

Expanding Power of Power – The Long Way

Let’s quickly review what a Power is, and how to expand a number raised to a power.

Image Copyright 2013 by Passy’s World of Mathematics

Now let’s look at what we do to expand an Index item that is raised to a power.

Image Copyright 2013 by Passy’s World of Mathematics

We can use this same multi-step expansion method for algebra letter variables.

Image Copyright 2013 by Passy’s World of Mathematics

Expanding Exponents Using “Power of Power Rule”

Image Copyright 2013 by Passy’s World of Mathematics

Let’s look at how we can do our two previous examples much faster by using this “Power of Power Rule”.

Image Copyright 2013 by Passy’s World of Mathematics

Note that when using this rule, there must only be ONE item inside the brackets.

This item is either a Number raised to a power, or a letter variable raised to a power.

If there is more than one Base item in the brackets, we cannot use the Power of Power Rule.

(There is a Power of Products Expanding rule which we will cover in a later lesson for dealing with these type of expressions).

Image Copyright 2013 by Passy’s World of Mathematics

Powers of Powers Videos

Here is a short simple video about the Power of Power Rule for indices and exponents.

In the following video the teacher develops the Power of Power Rule and then applies it to some expressions.

Power of Power Multiplications

The following examples show how to do Power of Power combined with multiplying terms.

These are basically a “two in one” expansion.

First we apply the Power of Power Rule, and then after this we apply the Multiply Rule.

Image Copyright 2013 by Passy’s World of Mathematics

Image Copyright 2013 by Passy’s World of Mathematics

This final Multiplication example shows that we need to remember that we can only add indices in the second step, if both bases, (the “Big” numbers), are exactly the same.

Image Copyright 2013 by Passy’s World of Mathematics

Power of Power Divisions

The following examples show how to do Power of Power combined with dividing terms.

These are basically a “two in one” expansion.

First we apply the Power of Power Rule, and then after this we apply the Dividing Rule of subtracting exponents.

Image Copyright 2013 by Passy’s World of Mathematics

Image Copyright 2013 by Passy’s World of Mathematics

Image Copyright 2013 by Passy’s World of Mathematics

Power of Power Multiple Choice Game

The above free online multiple choice game will help you get better at Power of Power Expansions.

Click the link below to play this game.

http://www.slidermath.com/wjava/Expnt5.shtml

Related Items

Subscribe

If you enjoyed this lesson, why not get a free subscription to our website.

To find out exactly how free subscription works, click the following link:

How Free Subscription Works

If you would like to submit an idea for an article, or be a guest writer on our website, then please email us at the hotmail address shown in the right hand side bar of this page.

If you are a subscriber to Passy’s World of Mathematics, and would like to receive a free PowerPoint version of this lesson, that is 100% free to you as a Subscriber, then email us at the following address:

Please state in your email that you wish to obtain the free subscriber copy of the “Power of Power Exponents” Powerpoint.

Help Passy’s World Grow

Each day Passy’s World provides hundreds of people with mathematics lessons free of charge.

Help us to maintain this free service and keep it growing.

Donate any amount from \$2 upwards through PayPal by clicking the PayPal image below. Thank you!

PayPal does accept Credit Cards, but you will have to supply an email address and password so that PayPal can create a PayPal account for you to process the transaction through. There will be no processing fee charged to you by this action, as PayPal deducts a fee from your donation before it reaches Passy’s World.

Enjoy,
Passy