Passy's World of Mathematics

Expanding Exponent Quotients

Posted on May 17, 2013 by Passy

Image Source: http://www.philipvaughan.net

Philip Vaughan is an amazing American Artist who Paints and Designs sculptures for public places.

He describes his work as “hidden geometry”, and in the picture above he has expanded out metal into spheres by making many geometric fractions which he has joined together and then added specialised lighting.

This lesson on Expanding Quotients involves fractions, dividing and expanding, but in a slightly different form to Philip’s work.

The word “Quotient” means one value divided by another value.

The Quotient expression can be written as either a Fraction, or as one item divided by another using a “divided by” sign.

Review of Expanding Products Rule

When expanding exponent products, we need to apply the power outside the brackets, onto every item inside the brackets.

The “Expanding Quotients Rule” is basically the same as the Products Rule, except that we are distributing the outside brackets power onto all items that are present in a Fraction.

Expanding Quotients – The Long Way

In the above examples we needed to multiply out the Top and Bottom of the given fraction three times.

We multiplied three times because we had a power of three outside the brackets.

If we look carefully we can see a pattern occurred from intitial brackets term, to the final expanded fraction without brackets.

The pattern, or shortcut, we can use is to apply the power outside the brackets, onto every item inside the brackets.

This shortcut is called the “Expanding Quotients” Rule, or the “Power of a Quotient” Rule.

Expanding Exponent Quotients Rule

Expanding Quotients Videos

The following videos explain the Expanding Quotients Rule, and show some examples of how to apply it to Bracketed Exponent Expressions.

The following excellent video from “Math Scribble” is about normal dividing of exponents; then 1 minute 42 seconds into the video covers
Power of a Quotient Rule.

The following video reviews the Products Rule, then at 1 minute 45 seconds into the video covers the Expanding Quotients Rule.

Expanding Quotients – EXAMPLES

The following examples show how to apply the Expanding Quotients Rule to both numbers and letter variables in bracketed expressions.

Because the expanded numbers would create a large numerical answer, we usually leave the answer in Index Form, rather than working out the actual number value.

Quotient Rule Backwards for Simplifying

We can use the reverse of the Quotient Rule to combine two different bases, THAT ARE BOTH TO THE SAME POWER, into a single bracketed item.

The two different bases can be: two numbers, a number and a letter, or two letters; however it is required that both items are raised to the exact same Power.

Quotient Rule Backwards – EXAMPLES

Using the Quotient Rule backwards is fairly easy, just look for a Fraction that has the Top and Bottom items both raised to the exact same power.

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Exponents in the Real World

Posted on May 17, 2013 by Passy

Exponents, Index Numbers, Powers, and Indices are used in lots of parts of our modern technological world.

Exponents are used in Computer Game Physics, pH and Richter Measuring Scales, Science, Engineering, Economics, Accounting, Finance, and many other disciplines.

Exponential Growth is a critically important aspect of Finance, Demographics, Biology, Economics, Resources, Electronics and many other areas.

Exponential Decay is associated with Light, Sound, Sporting Fixtures, Dangerous Chemicals, and Radioactive Waste.

People who use Exponents are Economists, Bankers, Financial Advisors, Insurance Risk Assessors, Biologists, Engineers, Computer Programmers, Chemists, Physicists, Geographers, Sound Engineers, Statisticians, Mathematicians, Geologists and many other professions.

In this lesson we show several Real Life uses of Exponents, as well as their impact on our understanding of the modern world around us.

Exponents are fundamental, especially in Base 2 and Base 16, as well as in Physics and Electronics formulas involved in Computing.

There has been an Exponential increase in the speed and power of computers over recent years, and by around 2030 computing power is predicted to match that of the human brain.

Exponents are critcally important in modern Internet based Sales and Marketing,

Exponents are important in Investing and Finance.

Compound Interest also works against people with a Credit Card debt they do not pay off, because the debt grows faster and faster each billing period and can quickly become out of control.

Exponents are the basis of “Demographics” (Population Growth)

The World’s Population is increasing at an extraordinary rate, especially in the devloping regions of Africa, India, and China.

With massive Population Growth comes massive use of Fossil Fuels for Industry, Heating, Electricity, and Transportation.

Over the last few years there has been massive exponential increases in mobile phone usage and market penetration.

Consumer Credit Debt has increased over recent years to record high levels.

Exponents are also part of Food Technology and Microbiology.

Virus Illness, (as well as many email and computer viruses), can spread at ever increasing rates causing major widespread infected areas.

This happens the same way that Viral Marketing branches out in ever increasingly wide branches of more and more people passing something onto more and more other people.

In explosions we get an uncontrolled massively increasing output of energy and force within a very short time period.

Picture this as a very steep exponential graph, compared to a burning match giving out energy in a fairly flat straight line graph.

Exponential Growth

The situations we have been considering so far involve “Exponential Growth”.

The equations for graphs of these situations contain exponents, and this results in the graph starting off slow, but then increasing very rapidly.

Eg. Think of Square Numbers and how they quickly get bigger and bigger:

1 4 9 16 25 36 49 64 81 100 121 132 etc

It only takes us nine square numbers to reach 100.

Exponential Growth situations when graphed look like the diagram below.

The opposite of “Exponential Growth”, is when we apply exponents to fractions which results in “Exponential Decay”.

Exponential Decay

Using negative power values results in fractions, and when these fractions have exponents applied to them we get “Decay”.

In a “Decay” process the amount involved drops off fairly quickly at the start, but then the drop off becomes slower and slower.

A typical Exponential Decay graph looks like this:

Making an Exponential Decay Graph

Image Source: http://teachers.egfi-k12.org

A fun way to make an Exponential decay graph is to take a pack of M&M’s or Skittles and keep pouring them out of a cup, but each time removing any candies which land with the letter side showing.

This should produce the required graph.

There is a great set of instructions on how to do this at the following link:

Click Here for M&Ms Exponential Graph Instructions

Exponential Decay – Real Life Examples

Some examples of Exponential Decay in the real world are the following.

Exponential Decay and Half Life

Many harmful materials, especially radioactive waste, take a very long time to break down to safe levels in the environment.

This is because these materials undergo exponential decay, and even a small amount of the material still remaining can be harmful.

Exponential Scales

The Richter Scale is used to measure how powerful earthquakes are.

The actual energy from each quake is a power of 10, but on the scale we simply take the index value of 1, 2, 3, 4, etc rather than the full exponent quantity.

This means that a Richter Scale 6 earthquake is actually 10 times stronger than a Richter Scale 5 quake. (Eg. 1000000 vs 100000).

Likewise, a Richter Scale 7 earthquake is actually 100 times stronger than a Richter Scale 5 quake. (Eg. 10000000 vs 100000).

The pH Scale for measuring the Acidity of materials is also created by taking the Power Values from measured powers of 10 acid concentration values.

Exponents and Scientific Notation

Very large numbers, like the distance between planets, or the population of countries, are expressed using powers of 10 in a format called “Scientific Notation”.

Scientific Notation is also used for expressing very small decimal values like the size of flu virus molecules, or the distance between atoms in a crystal structure.

Online Presentation on Exponents in the Real World

An online presentation of this lesson is available on SlideShare at the following Link:

Click here for our SlideShare Presentation

Music Video About Exponents

The following music video all about Exponents, is possibly the most successful Math video ever uploaded to YouTube.

It has currently had over 850,000 views on YouTube and is quite an amazing Production!

Well worthy of viewing by anyone learning Indices and Exponents.

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Expanding Exponent Products

Posted on May 16, 2013 by Passy

Image Source: http://upload.wikimedia.org

Hot air Balloons expand out to full size by filling them up with heated air.

In this lesson we are doing a different type of expanding out; one which involves two or more terms inside brackets that are raised to a power.

For two or more items inside brackets, that are then raised to a Power, we use the “Expanding Exponent Products Rule”.

This rule is kind of like a beefed up version of the Power of Power Rule.

At Passy’s World, we like to think of it as the Big Brother of the Power of Power Rule.

The Power of Power rule is used for when there is only one base item in the brackets, and works like this.

Expanding Exponent Products Rule

In the following example we look at a two items in the brackets expansion, the “long way”, and then work it out using the Expanding Products Rule.

Basically the Products Rule is as follows:

The Power outside the brackets needs to be applied to all Bases Inside the brackets.

Unlike other exponent rules, it is quite okay for these two bases to be different.

It is a lot like the Distributive Law, but is for Exponents.

It is important not to confuse the Products Rule with the Power of Power Rule.

The Products Rule applies when we have more than one Base Item in the brackets.

If there is only one Base item in the brackets, we use the Power of Power Rule.

The Exponents Products Rule can look a lot like the Power of Power Rule, but always has two or more items in the brackets.

It is kind of like the Power of Power Rule’s Big Brother, as shown in the following example.

Videos About the Exponent Products Rule

The following video shows the pattern whcih creates the Products Rule, and then does several examples.

The following video reviews the normal multiplication add rule, then covers Power of Power, and then at 4 minutes 28 seconds into the video, the Products Rule is demonstrated.

The following video also reviews the normal multiplication add rule, then covers Power of Power, and then around 5 minutes into the video, the Products Rule is demonstrated.

The following video shows how we can derive the Products Rule.

Expanding Exponent Products Rule – Examples

The following examples show how to apply the Expanding Exponents Product Rule when we have two or more base items in brackets raised to a power.

Product Rule Backwards for Simplifying

We can use the reverse of the product rule to combine two different bases, THAT ARE BOTH TO THE SAME POWER, into a single bracketed item.

The two different bases can be: two numbers, a number and a letter, or two letters; however it is required that both items are raised to the exact same Power.

Product Rule Backwards – EXAMPLES

We can use the reverse of the product rule to combine two different bases, THAT ARE BOTH TO THE SAME POWER, into a single bracketed item.

The two different bases can be: two numbers, a number and a letter, or two letters.

Exponent of Products Rule – SUMMARY

The following diagram summarises the forwards and backwards use of the Expanding Products Rule.

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Posted in Algebra, Exponents | Tagged Algebra, distributing powers onto ptoducts, expanding exponent products, Expanding Exponent Products Rule, expanding index products, Expanding Powers Products Rule, exponents, exponents and products, exponents index rule, how to do algebra with powers, how to do double exponents, how to do exponent powers, index laws, index rules, indices, power of power rule, power of products rule, powers, powers and products, product exponents, product of powers, product of powers rule, products index rule, raising powers to powers | 4 Comments

Power of Power Rule for Exponents

Posted on May 15, 2013 by Passy

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.

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.

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

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

Expanding Exponents Using “Power of Power Rule”

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

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).

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.

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.

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.

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

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Posted in Algebra, Exponents | Tagged Algebra, dividing Exponents, expanding double powers, expanding exponents, expanding powers, exponential growth, exponential population growth, exponents, exponents and population growth, exponents help, exponents power of power rule, exponents power rule, how to do exponents, how to expand exponents, how to use the power rule, indices, indices power of power, indices power rule, multiplyinging exponents, power of power rule expanding, powers | 5 Comments

Dividing Exponents

Posted on May 13, 2013 by Passy

Image Source: http://www.mycompasstest.com

In this lesson we look at how to Divide Algebra Expressions which contain exponents.

We also introduce the Dividing Exponents “Subtraction” short cut rule.

If you are very new to Exponents, we recommend that you first do our previous Exponents Lessons, which are accessable at the following links:

http://passyworldofmathematics.com/basic-exponents-and-indices/

http://passyworldofmathematics.com/multiplying-algebra-exponents/

http://passyworldofmathematics.com/algebra-dividing-and-exponents/

In our previous lesson on Algebra Dividing, we looked at dividing exponent terms by fully expanding out and cancelling identical terms, as in this example:

Exponents “Subtract Powers” Dividing Rule

Rather than do lengthy expansions and cancelling out, we can use a shortcut rule for doing our Exponent Division.

This shortcut rule is similar to the “Add Powers Rule” which we have learned previously for Multiplying Exponents.

The following example shows how to do our previous question using the Dividing – Subtract Powers Rule.

The Subtract Rule also works for Expressions containing letter variables.

Videos about Dividing Exponents

The following quick video shows how to do dividing that involves exponents, using the subtraction rule.

The following video shows three examples of dividing exponents.

Steps for Dividing Exponents

For the more involved expressions we will be doing in the remainder of this lesson, the following working out steps need to be followed:

Dividing Exponents Examples

The following example shows how to break apart and simplify a multiple items exponents division fraction.

This next example shows how to do an exponents division which involves negative numbers.

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Posted in Algebra, Exponents | Tagged Algebra, algebra fractions, algebra quotient rule, dividing algebra, exponent fractions, exponents, exponents and fractions, exponents quotient rule, exponents subtract rule, how to divide exponents, how to divide indices, how to divide powers, how to do exponent fractions, indices, indices and fractions, indices quotient rule, powers, powers and fractions, powers quotient rule, powers subtract rule, Quotient Rule, simplifying exponent fractions, simplifying Indices | 5 Comments

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