Passy's World of Mathematics

Algebra Dividing and Exponents

Posted on May 15, 2012 by Passy

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In this lesson we cover dividing algebraic expressions.

It is important that you have some prior background knowledge in Algebra and Exponents.

If you are unsure, then check out our previous lessons on this material at the links below:

http://passyworldofmathematics.com/multiplying-basic-alegebra-terms/

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

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

It is also important to remember the Integer Rules for Dividing positive and negative numbers.

These dividing rules can be summarised as follows:

Finally, you will also need to be familiar with breaking down numbers into their Prime Factors.

Watch the videos in the following section if you do not already know how to make “Prime Factors” for numbers.

Prime Factors Videos

And finally watch this great music song video from “Math Rocks!”

Video About Algebra Division

The following video does get into some advanced bracket divisions, but does include some simpler examples related to the work we are covering in this lesson.

Basic Algebra Division

Let’s first do a normal division of 12 divided by 6, which we should all know equals two.

In the example below, we show how to do this division using the cancelling prime factors method.

This may seem like a very long way to do a simple division; however this cancelling factors method of reducing down fractions will be very useful when we need to reduce down Algebra Fractions.

Let’s start our Algebra Divisions with something very simple : 12w / 6

If we separate the numbers and the letters we have 12/6 times w.

12 / 6 means 12 divided by 6, which is 2.

So for our answer we have 2 times w, which is written as 2w.

Algebra Division Steps

The steps for working out any Algebra Division question are as follows:

Algebra Division Examples

The following examples show how to do a variety of Algebra Division questions, including ones which contain Powers Terms with Exponents.

Video About Multiplying and Dividing Exponents

This final video revises multiplying exponents, and then later on gets into dividing exponents.

It also shows the fast track rules of adding Exponents for Multiplying Problems, and Subtracting Exponents for Dividing Problems.

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Multiplying Algebra Exponents

Posted on May 14, 2012 by Passy

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Computer Games use “Game Physics Engines” which are low level programs inside the game to calculate the movement, interactions, and the geometry involved with the game.

These programs use lots of Algebra formulas in their Algorithms, and many of these formulas involve multiplying powers terms containing exponents.

If the mathematics isn’t correct in the game engine, then the game is not going to play at all like we would expect it to.

Revision of Previous Lessons

Before doing today’s lesson on “Multiplying Algebra Exponents” you need to make sure you understand some previous background work we have done in Algebra.

Check out the previous lesson links below for material on basic exponents and multiplication, and make sure you are familiar with what has been covered in these lessons.

http://passyworldofmathematics.com/multiplying-basic-alegebra-terms/

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

http://passyworldofmathematics.com/algebra-powers-substitution/

http://passyworldofmathematics.com/multiplying-integers/

The most important previous work we will be using for our Algebra Multiplying involves the expansion and simplification of Powers.

Add Exponents When Multiplying Rule

Consider the expression 2 cubed times 2 to the power of 4.

We can expand the exponents and then work out a simplified answer.

We can also multiply and simply Algebra exponents.

There is a simple pattern that is happening here.

The final value of the exponent is always the added up total of the exponent values on the original bases.

This means we can use the following shortcut rule for multiplying exponents.

However, we need to be very careful when doing exponent multiplications.

Not all questions can have the Exponents Adding Rule applied to them.

Some questions have exponent terms which have different bases, and these questions do not involve using the shortcut adding rule.

Instead they are done by following the normal Algebra Multiplying process.

Remember to check very carefully that the big number or letter bases are the same before applying the “fast track” adding rule to multiplication problems.

Videos About Multiplying Exponents Rule

The following video is quite long and gets into working out some fairly involved questions towards the end, but is well worth watching as it covers everthing we need to know about Algebra Multiplication of Exponents.

Here is a video which summarises the main types of exponent multiplication questions.

This next video covers the same type of material as the other videos already included in this lesson.

Finally, if you would like to watch one more good video about Exponents Multiplication, then click the following link.

http://www.youtube.com/watch?v=cV3xWPyZpQI

Steps for Multiplying Algebra Terms

Remember to always check carefully that the bases are the same before applying the fast track adding rule to any terms in questions.

Multiplying Algebra Terms Examples

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Posted in Algebra, Exponents | Tagged adding with multiply exponents, algebra exponents, algebra help, algebra lessons, algebra multiplication, algebra multiplying, algebra powers, algebra simplification, algebraic multiplication, algebraic simplification, exponents math, exponents multiplying, exponents rules, help for algebra, help in algebra, help on algebra, help with algebra, how do you multiply, how to algebra, how to algebra multiply, how to multiplication, how to multiply in algebra, how to multiply variables, indices multiply rule, indices rules, math variables, math with variables, monomials multiplication, multiply exponents, multiply with exponents, multiplying Exponents, multiplying indices, multiplying monomials, multiplying rule, multiplying rule for exponents, multiplying variables, multiplying variables in math, multiplying with indices, rule of adding exponents when multiplying, simplify algebra expressions, simplify exponents, simplifying exponents, simplifying Indices, simplifying powers, simplifying with exponents, variables in math | 9 Comments

Multiplying Basic Algebra Terms

Posted on May 12, 2012 by Passy

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Basic Multiplication

The following example shows the simplest Algebraic Multiplication.

In this example we have to increase “3b” by a factor of 4.

Algebra Multiplication

Now let’s consider the multiplication of 4a x 3b or 4 Apples x 3 Bananas

This is really strange to multiply Apples by Bananas because they are different items !

Let’s do it anyway ! First let’s put in all the multiply signs:

4 x a x 3 x b (Now adjust the order so the numbers come first)

In multiplying we are allowed to rearrange the order because it does not change the answer.

Eg. 5x2x3 = 30 and 2x5x3 = 30 and 5x3x2 = 30 and so on.

Mathematicians call this the “Commutative Property” : Factors can be multiplied in any order and the product is always the same.

So let us rearrange the order of our apples and bananas multiplying:

4 x a x 3 x b

= 4 x 3 x a x b (We can work out the number part as 4 x 3 =12)

= 12 x a x b

= 12ab (Remember in Algebra we leave out x signs from answers)

Now this is really hard to get your head around, but 12AB could be the number of Grapes that we have.

In this case we have lots of Grapes!

The Total Grapes = 12 x the number of Apples x the number of Bananas.

So 4a x 3b is an expanded way of saying that we have 12 x a x b or 12ab.

Practical Application of Algebra Multiplying

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Hopefully the following will help you understand one real world use of Algebra Multiplication.

Algebra multiplication equations are useful for Restaurants and the Hospitality Industry, where catering is done in large dining rooms.

It would be a major fail to set up a function room or restaurant and not have enough knives and forks for all of the guests to be able to eat!

Real World Example

For a function we might have “T” tables that each seat “N” people, where N could be 6, 8, 10, or 12 depending on the function room.

Each person at a table requires two big forks, (one for eating the entree and one for the main course).

The total number of forks required for setting all the tables at the function is:

F = 2 x N x T = 2NT

In the good old days, Algebra Calculations like this would be worked out on paper, but these days there is probably a mobile device or computer app which can do most of the work for us.

We would simply enter:

Number of Tables

Maximum Number of people at each table

Number of big forks needed per person

Number of little forks needed per person

Number of big knives needed per person

Number of little knives needed per person

Number of spoons needed per person

Number of teaspoons needed per person

and the App would calculate our total cutlery requirements.

The Algebra is programmed into the App by the person who wrote the App.

Algebra is essential for programming any computer device, and a valuable skill to have in these modern times.

Steps for Algebra Multiplication

The steps for carrying out Basic Algebra Multiplication are as follows.

Integers Multiplying Rules

For Multiplying integers, we use the following set of rules.

If you need to review Multiplying Integers, then click the following link:

http://passyworldofmathematics.com/multiplying-integers/

Also it is important to remember for this lesson that:

b is the same as 1b, and -k is the same as -1k etc

Scope of This Lesson

Algebra Multiplication can involve Exponents, Indices, Powers, and multiple terms inside brackets.

The lesson we are doing here is an introduction to Algebra Multiplication and only covers beginner’s basics.

Our examples will only be for very simple “monomials” and “terms”.

More advanced Algebraic Multiplication involving Powers and Brackets will be covered in separate lessons.

Algebra Multiplying Examples

The following examples show how to do some typical Algebraic Multiplication and simplification questions.

It is important to remember that for negative Variable letters without coefficients, we do the following:

-y is the same as -1y, so we re-write -y as -1y when multiplying

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Posted in Algebra | Tagged algebra help, algebra lessons, algebra multiplication, algebra multiplying, algebra simplification, algebraic multiplication, algebraic simplification, help for algebra, help in algebra, help on algebra, help with algebra, how do you multiply, how to algebra, how to algebra multiply, how to multiplication, how to multiply in algebra, how to multiply variables, math variables, math with variables, monomials multiplication, multiplying monomials, multiplying variables, multiplying variables in math, simplify algebra expressions, variables in math | 3 Comments

Combining Like Terms

Posted on May 11, 2012 by Passy

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People who are twins are a “Like Pair” because they have the exact same features.

An Algebra Equation we can write for this is :

T + T = 2T

Eg. One Twin + One Twin = 2 Twins.

In Algebra, when there are items that are the same thing, we can add them together to make a simpler expression.

Introductory Videos About Like Terms

The following is a video taken of a PowerPoint presentation and has no sound, but does a great job of introducing the concept of “like terms”

Here is how a typical Like Terms Algebra problem is done:

Definition of Like Terms

“Like Terms” are terms that contain the same letter Variables raised to the exact same Powers.

( Only the first number “Coefficients” of the like terms are different).

3a and 2a are like terms because although they have different coefficient numbers, they have the exact same letter “a” in them.

The letters and powers in the items are always identical whenever we have “Like Terms”.

It is extremely important that the Exponent Index power values are identical on each letter.

If they are not identical, then we do not have Algebraic Like Terms.

Identifying Like Terms

It is important to fully examine each term very carefully.

Just like two cars might be nearly identical, we need to examine the two cars fully to notice that they are not exactly the same.

Here are some examples where you need to decide if the given items are “Like Terms”.

Here are the answers to the previous examples.

Why We Combine Like Terms

Often in real life it is necessary to combine like items together to create a shorter list of items we can deal with.

For example, imagine that a mathematics class is on an excursion and need to order a take away food lunch.

It would be crazy to read out each individual order, one after each other, at the counter of the fast food restaurant.

Instead we would total up how many burgers, how many fries, how many drinks, etc that we needed for the whole group.

This type of summarizing process is exactly the same as combining like terms.

The following video might get really annoying after a while, but it definitely shows why we need to group items together into separate sub totals of each type when doing mathematics.

How to Combine Like Terms

We can add together items that are the same to make a simplified shorter list of items.

This is called “combining like terms” or “collecting like terms”.

Consider the following family take away order:

Two burgers, one fries, one drink, three more burgers, two more fries, and two more drinks.

This order needs to have the like items grouped together to make a shorter summarised list.

Subtracting Like Terms

We can also subtract Like Terms.

The following Example shows how we do this.

WARNING: Like Terms are only used for Adding and Subtracting terms.

We never use combining like terms for Multiplying and Dividing !

Multiplying and Dividing use a totally different method, and they must not be done using the grouping Like Terms method.

Videos About Combining Like Terms

This video is about Combining Like Terms with Professor Perez and Charlie (7 mins 30 secs long), but well worth watching.

If you are not getting the idea of like terms yet, then watching this coloured shapes video should help.

Video Examples for Like Terms Questions

This video shows how to identify like terms, and also how to combine them to make simplified answers.

The following video shows how to do several simplifying like terms questions.

Steps for Combining Like Terms

The steps for combining like terms are as follows.

Like Terms Examples

The following examples show how to do some typical Like Terms simplification questions.

Matching Like Terms Game

In this game check what the “Target” value is, and then click on all squares that are Like Terms as fast as you can.

In the screen print above, the Target is a number, and so we need to click on all squares that contain ordinary numbers in them with no letters present.

http://themathgames.com/our-games/like-terms-games/play-matching-action/

Like Terms Invaders Game

In this game you need to shoot at aliens that have the same term as on your bottom of screen shooter.

However, you also need to keep using the mouse to move and dodge Alien rockets firing at you.

Use the left button on the mouse to shoot.

Click the link below to play this fun game.

http://www.xpmath.com/forums/arcade.php?do=play&gameid=92

Combining Like Terms Millionaire Game

Click the link below to go to this game, and then click the “Play Game” box.

http://www.quia.com/rr/195128.html

Like Terms Online Quiz

Click the link below to go to an Online Quiz on Like Terms.

Under each question, click on the blank line and a box will appear where you can type in your answer.

At the end of the quiz, you can click the “How did I do Charlie” button to check your answers.

http://www.saddleback.edu/faculty/lperez/algebra2go/prealgebra/variables/combLikeTerms.html

For more great lessons, videos and quizzes on Algebra, check out the “Algebra2go” website at this link:

http://www.saddleback.edu/faculty/lperez/algebra2go/

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Posted in Algebra | Tagged adding and subtracting like terms, adding like terms, adding like terms in real life, algebra expressions, algebra like terms worksheet, algebra terms and definitions, algebraic expressions equations, algebraic problems, algebraic term, basics of algebra, collect like terms, collecting like terms, combine like terms, combining like terms, definition of like terms, help for algebra, help in algebra, help on algebra, help with algebra, how do you combine like terms, how to combine like terms, how to simplify equations, like terms algebra, like terms definition, like terms worksheet, real life adding like terms, real world like terms, simplification of algebraic expressions, simplifying like terms, subtracting like terms, what are like terms, what is a like term | 3 Comments

Algebra Powers Substitution

Posted on May 4, 2012 by Passy

Image Source: http://resources0.news.com.au

This car contains “Powers” – the power of a very hi-tech engine, as well as racing car driver “Will Power”.

When we substitute plenty of raw power into a racing car,including racing driver Will Power, we get some very large speeds produced around the track !

When we substitute numbers into Algebra Expressions that contain powers, we get a large number of multiplications to work out.

If you are not familiar with Indices, Exponents, and Powers, then the following lesson is compulsory preparation for what we are about to do here:

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

Working Out Steps for Powers

Here is a typical Powers substitution question:

The steps to working out all Powers substitution questions are as follows:

Powers need to be expanded using the basic exponents rule of multiplying the base out as many times as the small Index number says to do.

We also need to apply the Integers Operations Rules that are needed for multiplying and dividing positive and negative numbers.

Finally we also need to do our working out following the standard “Order of Operations”

Powers Substitution Video

Watch the following video which shows some typical substitution questions whcih involve exponents.

Powers Substitution Examples

The following examples show how to do some typical Powers Substitution questions.

Practical Example of Exponents Algebra

Algebra Formulas are used in the real world to work out the volume of various shapes.

Cylinders are a critical part of car engines, and the size of the motor is specified by the total volume of all of its cylinders.

Mechanics and Engineers need to know how to use Algebra Formulas.

The volume of each individual cylinder is calculated using the following Algebra formula:

For example a car engine might be a 3.8 litre V6, or a 400 cubic inch V8.

The following example shows how to calculate the volume of a car cylinder, using Algebra substitution.

Values Table Substitution

We sometimes need to use a supplied rule equation to complete a values table of “x” and “y” values.

These tables are used when we need to draw a graph for the rule. However we will not be drawing the graph in this example.

This next example is basically four separate substitution questions, where we are putting our four answers into an answer table.

We now work out “y” answers for the other three “x” values the same way.

At Passy’s World we find it easiest to do these questions by adding a middle column to the table where we do the working out.

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Passy

Posted in Algebra, Exponents | Tagged Algebra, algebra exponents, algebra help, exponents, exponents games, exponents help, exponents worksheets, free math lessons, how to do exponents, how to do indices, how to do powers, index numbers, indices, introduction to exponents, math help, mathematics help, mathematics lessons, powers, writing exponents | 5 Comments

Passy's World of Mathematics

Algebra Dividing and Exponents

Multiplying Algebra Exponents

Multiplying Basic Algebra Terms

Combining Like Terms

Algebra Powers Substitution

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