# Mathematical Literacy

## Integrating Literacy Development Opportunities in Your Instruction

A few years ago I attended a professional development seminar designed to help American math teachers integrate best practices and strategies required for their students to be successful with the program.  I was a guest “reference-source,” in the seminar because of the success my students experienced in the program over the prior six years.

The IGCSE program is, in short, a college preparatory program.  By passing the end of course examinations students can demonstrate college readiness.  In my school they’re even given a high school diploma at the end of their 10th grade year, upon successful completion of the program of course.  Some have even exited the school to attend college during what should’ve been their 11th grade year.

At the end of the seminar participants were invited to ask questions.  A teacher, quite frustrated, asked, “How am I supposed to get my freshmen prepared for calculus by their senior year?  There are too many things to teach and not enough time.”  (What she was getting at is that the Cambridge curriculum is appears sparse compared to typical American curricula.  In 9th and 10th grade there are a total of 10 topics for math.)

The presenter asked me to handle the question.  I knew the answer, but could not articulate my thinking in a concise fashion.  She and I were speaking different languages.  I tried to explain that she didn’t have to teach everything.  It is better to have a solid foundation that can be applied to all of the tangential and “one-off” topics in math, than it is to have brief experience with all of those various topics.  We do not have time for both.  We cannot develop deep understanding of the fundamentals of Algebra and have students exposed to all of the iterations and applications.

All she heard was a know-it-all teacher bloviate about some theoretical ideal.  She needed practical advice.  While I tried to provide that advice, I failed, miserably, to do so.  I realized after writing this article that this information, in this article, is what I should have shared with that teacher.

Her question had a specific context, but I believe it hit the heart of one of the biggest issues faced by mathematics teachers, world-wide.

#### How do I get my students to acquire and retain mathematical thinking?

I’m going to offer a two-word solution:  Mathematical Literacy.

If we want our students to really learn mathematics and be flexible enough to apply that knowledge in their futures, they have to be mathematically literate.   Mathematical literacy, for our purposes here, is (1) the ability to decode information from mathematical text and (2) the ability to encode contextually relevant information in mathematical text.

A mathematically literate person can understand mathematics as it is written, but also realizes countless associations, contextual meanings, and tangential ideas.  When a mathematically literate person sees mathematical text, they don’t wonder what should be done.  They read it as information, which is decoded and analyzed.  They are able to articulate appropriate, contextually relevant mathematical responses to information provided.  A student that has developed this literacy is prepared for whatever type of math their futures may hold.  They’re not bound by our efforts, they’re not reliant on what we have directly shown them.  Instead, they’re empowered with the ability to think and communicate mathematically.

The prior two paragraphs are entirely insufficient for defining mathematical literacy.  This article is about developing mathematical literacy, not defining it.  If you’re interested in learning more about what is meant by mathematical literacy, consider listening to the On Teaching Math podcast on mathematical literacy.   You can access the podcast with this link.

The development of mathematically literate students involves two components.  First, students must make sense of problems they’ve never seen, and problems that often expose a misconception created by a person overly reliant on procedural proficiency.   Then, students must apply something they know that is contextually relevant to the problem at hand.  The key component here is they must identify the relevant concept and understanding.  They must make the association.  They cannot be following a mapped-out procedure or following instructions.

In order for this to happen, students must have a certain degree of conceptual understanding and procedural proficiency.  However, marginal proficiency with both is sufficient.  Through developing literacy, they will improve their conceptual understand and their procedural proficiency.

Warning:  Carefully acclimate students to answering questions designed to improve literacy.  If you do the thinking, instead of teasing it out from them, you can destroy the possibility of developing literacy in students.  As we dive into a few examples we will discuss, in detail, how this works.  But, for now, understand that if you demonstrate how to solve the problem, or answer the question, students will not develop literacy.  In order to develop literacy, students need to bring in relevant conceptual understanding (may or may not be directly related to the topic being taught), and then devise a plan and monitor the appropriateness of their approach as they work through it.

If we, the teachers, make all of the connections and do all of the decision making, we’re the ones exercising our own literacy.  Literacy will not be developed through imitation.

Let’s get into how we can set up experiences for our students that will promote the development of mathematical literacy.  We will use solving simple polynomial equations in Algebra 1 as our initial  testing ground.  View these examples in their spirit, not specific application so that you can begin to craft your own questions and design their implementation.

Suppose your students have been taught how to combine like terms, and then solve simple equations, like 3x + 3 + 4x – 5  = 23.  You can run them through countless pages of practice where they’d see every possible iteration of this type of problem.  But, you’d not be increasing their literacy or developing a deeper conceptual understanding.  That would only promote procedural proficiency, which is of course not well retained over time.

Instead, you could give students a problem like Problem A.

The problem on its own will not promote literacy.  How you introduce the problem and your expectations of students will promote mathematical literacy.  If you work a similar problem, by changing the numbers, the students will latch onto the procedure.  They will not be pulling in various mathematical understandings they possess that are contextually relevant.

However, without support, at least initially, students will likely be unable to even approach this type of problem.   The level of performance and thinking required of your students is likely brand new, and foreign to your students.  They will wait for you to show them how it goes, and then try to recreate what they’ve witnessed.  That is exactly what we do not want.

If this was the first opportunity for my students to develop mathematical literacy, I’d explain my expectation and goal to them first.  The purpose of the problem is not to find an answer, but to develop the ability to understand what is written and draw in previously held understanding.  Once the understanding and associations are complete, students are practicing articulating their thinking mathematically.

The purpose of this problem is to provide students with experiences that will prepare them for unknown futures.  This is practice that will help make them adaptable by teaching them how to think mathematically.

A good way to start is to show students the diagram and the information, but not the question.  Ask students to brain storm about what they see, what they know, what comes to mind.  They’ll often be hesitant to state the obvious things, but those obvious things are sometimes the most difficult to see and are sometimes the most important things to notice!

Once students have collaborated, through whole-class discussion collect and list ideas and observations on the board next to the diagram.  Many kids will have forgotten how perimeter works.  This will be a great time to shore-up that issue.

Then, after all of the observations have been recorded and discussed, show students the question.  Remind them that the steps to be followed are not what is important here.  Creating the steps to be followed is what’s important.   We want students to write mathematically, in appropriate contextual response to information provided.

Unfortunately, once this introduction has been completed, the opportunities to develop literacy with this style of problem are long gone.  The road is familiar.   Students will be remembering the process instead of making mathematical connections.  In response to this, teachers need to have two things at the ready.

1. Students coached to fully engage with the problems.  They cannot sit back and wait for the path to be clear.  Finding the path amidst uncertainty is the pursuit.  Once a problem has been explored, the path is found and the goal is no longer attainable.
2. You need a bank of problems at the ready!

Here is another, similar, but fundamentally different problem that could be used to follow Problem A.

Of course helping students develop the habits of thinking that will lead to literacy takes time.  You could easily teach students to “do” this problem in a handful of minutes.  Then, you could try to back-fill some meaning.   But then, students are learning how to “do,” the problem.  They’re not getting practice learning how to thinking mathematically.

The pay-off, however, is worth the time spent!  By learning how to make sense of mathematical information, and how to identify contextually important prior knowledge, then articulating their thinking mathematically, students will, over time, learn much more quickly.  They will also strengthen the prior knowledge through these experiences because these experiences provide opportunity to create connections between topics.  All of these benefits together result in greater retention of the new, and old, mathematical concepts.

Let’s see an example that would be appropriate for students at this level that does not involve Geometry.  Again, we are considering a group of students who can distribute and combine like terms, and solve equations in one variable.

There are two boys, John and Bob.  Both boys like to collect colorful rocks.  Bob puts his rocks in his left pocket, which has a hole in it.  John finds half of the rocks that Bob drops.

If Bob found 36 total rocks, and one third fell out of his pocket, how many of Bob’s rocks did John find?

There is nothing special about this problem, or the previous two.  What is different is how you introduce the problems and how you coach students to approach the problems.  Encourage brainstorming, making sense of the problems.  Set the expectation that students will need to develop mathematical literacy in your class to be successful.  If it is a true expectation, and you are unwavering, but encouraging, students will develop literacy over time.

Questions that are not directly related to the topic at hand can also be used.  In my podcast, On Teaching Math, I start each episode off with a question like this.  They’re typically easily understood and involve solutions that are within reach of most people, regardless of mathematical prowess.  Also, it is often the case that the answer or discovery made by exploring the question is of little consequence.  But, what is important is that students must create a hypothesis and test it, either through independent exploration or collaboration.  As they test their hypothesis, through reflection they must decide to adjust their or approach, or through validation, continue on.

A typical question will be:  How many times in a 24-hour period will the hands of a clock create a 90-degree angle?

Another question that is simpler is: Why is 5 the only prime number that is the sum of the previous two prime numbers?

One more example is:  What number less than 100 has the greatest amount of unique prime factors?

These types of problems are a great way to give students experiences that develop mathematical literacy.  The way a person must engage with those problems is the same way a mathematically literate person can engage with our last example.

One last positive outcome from these questions is that a lot of meaning will be exposed. Students will likely discover things you never thought of.  That is a great outcome and a great way to include activities that promote academic discussion into your classroom.

This final example is a favorite question that can be used to develop literacy.  An ancillary benefit is realized for students who failed to obtain the solution.  In review, students will have a deeper understanding of exactly what the concept at hand with this topic really means.

Suppose you’ve taught your students the mechanics of functions.  They can read and perform operations from examining the notation, they can perform function arithmetic, maybe composition of functions, and they can find inverse functions.  I selected the words, can find, here because they indicate procedure, not concept!

Here’s the question:  Given that f(x) = 2x, what is the value of x when f -1(x) = 4?

When I first saw this question on a Cambridge IGCSE examination I thought the question was entirely unfair!  In fact, I was asked by a person outside of my district how kids could solve this.  The students taking the test had no experience with how to find the inverse of the function!

When the test results were released I was shocked to see that the majority of my students answered the question correctly.  I could not believe it.  Upon questioning, students explained that the question was easy because the input and output for a function and its inverse are reversed.  For example,  if g(2) = 3, then g-1(3) = 2.  So, if the output of the inverse of function f is four, then the input for the function f is four.  Then, f (4) = 24, which is 16.

Because the students understood the concept and had practice applying concepts in new ways, they were successfully able to answer a difficult question correctly!  To make it even better, they answered a question that I had never dreamt of before.  This is a great example of the power of mathematical literacy.

Let’s pull it all together here.  To develop mathematical literacy students must apply conceptual understanding in non-routine applications.  This will likely be a shift in engagement for students and teachers.  As such, we, the teachers, must orchestrate a series of experiences that will help students make this shift.  We start students off with simple to understand questions that are non-algorithmic in nature, and gradually move to more complicated application of the concepts at hand.  All the while, we increasingly move students to more independent thinking, where they collaborate AFTER they've have created and executed a plan. The pay-off is well worth the time and effort required!  This is absolutely a case where going slow early can speed things up over time!

Your devotion and consistent application are required to help students develop mathematical literacy.  You will need to incorporate these style of problems and the appropriate pedagogy into your lessons.  Students will need opportunities to practice their literacy on homework, quizzes and tests.  Many of the students will require continual encouragement and reiteration of the relevance of their efforts (why it is important for them, that they develop literacy).

If your students develop mathematical literacy under your tutelage, then you will have served the future needs of that student well.  They will be prepared for an unknown future because they will be empowered with the ability to think, and communicate, mathematically.

If you are looking for questions that can be used to promote mathematical literacy within the application of a specific topic in math, please leave me a comment below.  I have a large collection of these types of questions built over the years.

# My Favorite Technology

With the invention of the radio came claims that, “This will revolutionize education, forever.”

Then came television, and more claims that, “This will revolutionize education, forever.”

Then came the VHS player.  You guessed it, more, “This will revolutionize education, forever.”

Then the internet came along, and louder than ever were the claims that, “This will revolutionize education, forever.”

In truth, all of those pieces of technology have revolutionized education.  Education is now, more than ever, about coming up with new ways to make information increasingly accessible and more engaging.  And, more kids than ever are starting college.  What’s not to love, right?

Well, there is plenty not to love.

The reason all of those pieces of technology were destined to change education forever was because they were going to allow experts in particular fields to communicate with students.  The thinking was that books and stuffy teachers were making learning unnecessarily difficult.  By allowing students to bypass the texts and teachers to gain access to the content to be learned, they’d learn better and faster.

It makes sense to me.  As an adult, if I want to learn about writing a blog, for example, I do a search on the internet and find some self-proclaimed blog expert.  I watch their videos, read their blogs for advice, and give it a shot!  Or, if I want to learn to change the air filter on a new car, and I can’t seem to figure it out myself, I look for videos on YouTube.  Technology like the internet has provided me with so much greater access to information that has enriched my life than was afforded before the internet.

That’s how it is supposed to work with students, too.  A kid might be stuck in Algebra 2; logarithms killing my grade, mister!  They look up “logarithms,” on the internet and there are tons of helpful videos.  The student learns how to do logarithms, and their grade is saved.

It sure seems like it is all on the up and up, right? Well…

With up to 60% of college freshmen needing remedial math classes, I’d say these revolutions have not had a positive outcome for students.

At this point you might be thinking, here’s another doomsday message: Kids these days are horrible, fear for the future.  I promise you, this is not a doomsday message.  Education needs to improve, and that’s what this blog is about.

Let’s take a step back and look at the example where I learned to change an air filter from a YouTube video.  Was I educated?  Was I trained?  What’s the difference?

There is a huge difference between training and education.  Training equips the trained with specific skills and knowledge that the trainer knows the learner will need, when they will need it, and how they will apply what they’ve been trained to do.  Training is what happens when you get a new job.

Training could be said to equip a person with a specialized tool.

Education is different.  People often complain why they weren’t taught certain practical skills in school.  The message is that education is worthless.

Education equips a person with the ability to find the specialized tool they need and then figure out how to use it.  While training prepares someone for a known task, education prepares someone for an unknown task.

When a student watched a video on the internet about logarithms are they being trained or educated?

If the intent of the video is to help a student complete homework and pass a quiz, then the person knows exactly what the student will need to be able to do, and when they’ll need to do it.

This is a seemingly subtle difference.  The difference between training and education is anything but subtle.  It is of massive consequence.  Why?

One attribute of an educated person is that they quickly incorporate new, more effective approaches.  By contrast, a trained person resists new methods, regardless of efficacy.  Education makes a person adaptable.

The reason that the radio, television, videos, and the internet have failed to improve educational outcomes is because they have not addressed the short-comings of a textbook.  All of these sources provide the same information, and use the same approach.  They disseminate information.

A good teacher entices curiosity, finds what motivates students to learn, and provides educational experiences for students.  That quality human connection is what makes education happen for students that are otherwise uninterested in being educated (which is an overwhelming majority).

There is such a massive push, with some much inertia behind it, to focus on comprehensible input, scaffolding, all of the components of teaching examined in isolation and treated with a leaning towards training a teacher instead of educating them about teaching, that it feels like quality teaching is becoming a lost art.  Maybe that’s a skewed perspective having only taught in Arizona, which by nearly every metric, is the worst state for education in the US.

What students need is a reconnection with their instructor.  The instructor needs to get in-tune with the needs, pace, and interests of the students.  PowerPoints, videos, SmartBoards, Chrome Books, and the like focus on the dissemination of information.

That is why my favorite piece of technology is the document camera.

Wait … hear me out.  I believe that it can be the most powerful piece of technology for a student in a math class.

The first reason why I love the document camera deals with how mathematics is a written, not spoken language.  The spatial arrangement of characters conveys meaning.  The way math is printed on paper, or a PowerPoint, and the way it is written on the board, is different than how math is written and performed on paper.  The physical parameters change the way we write.

In the image below you can see a lot of repeated information.  Some of the information is written mathematically, some of it is written in English, and there are arrows and annotation that connects the two.  These annotations are done in real time in response to questions from students and answers by students to my questions.

The way we write math greatly impacts how we perform the math.  This is an overt example, but I think it will make the point.  The first expression below is extremely difficult to deal with, while the second has the same meaning and is quite easily understood.

A bright student might realize to rewrite the first expression as the second.  But an average student will realize with the second that they only need to add the exponents, and they’re done.  This is not an example of how the interaction of math is different when writing on paper, versus typing.  What it does show is that how math is written greatly impacts the interpretation of the meaning.  That interpretation and translation occurs more naturally when written in real time compared to being typed.

What the example above does show is how writing mathematics drastically changes our interpretation of what is written. In effect, it rephrases the information.

What the document camera does is allow the teacher to show students, in real time, the mechanics of the mathematics, while allowing for discussion and annotation of the theory of the mathematics.  It does these things at a writing pace.

Here is a picture of a lesson in Algebra 1.

In this picture what you see is how a problem can be broken apart in response to what it is that the students in the classroom, at that very moment in time, are struggling with.

While this could have been addressed while writing on the board, or even in a PowerPoint lesson, it was more apparent to me, as the teacher, because I was going slower.  I was asking more questions, students were asking more questions.  Teaching with the document camera really can improve the dialogue between teachers and students, changing it from speaking to conversing.

While a conversation can be had over a YouTube video with students, or during a PowerPoint presentation, it is more difficult.  The pace is different; the engagement of the students is different.  When watching a video, or watching a PowerPoint, students are … watching.  If they begin writing, it is often dictation that’s being performed.

There is certainly a measure of dictation happening by students when engaged in a lesson delivered through a document camera.  However, the switch to addressing a question or point of confusion during a lesson in a way that students incorporate that response as a natural part of the lesson, happens naturally when using a document camera.

Consider a lesson about exponents.  No matter your teaching experience, you cannot anticipate all possible misconceptions, prior or actively developing, and dispel them pre-emptively.  Along the way there will be confusion and misunderstanding.  It is when the confusion is discussed, and properly addressed, that learning really takes place.

When that confusion is brought forward by the students, in a lesson delivered through a document camera, the question can be written, explored, answered and summarized in a way that feels natural for the students.  They’ll recognize this as part of the lesson, not a tangent.

In the picture below you will see a refocusing of a concept learned the day before.  In the day before this lesson, students really struggled to identify separate bases in one expression.  They could not distinguish between things like

Of course confusion is exposed and can be properly addressed in other delivery forms.  The message here is not that other methods are ineffective.  However, students typically view a diversion from the script as tangential to the lesson objectives.  They do not recognize that the diversion is the most important part of the learning.  How could it be when it doesn’t have pretty animations and bold, underlined font?

The last benefit of a document camera is pacing.  Students need think time.  The pace of delivering a message is slowed when you, the teacher, are essentially taking notes with the students.

This allows them to think about what is being written while they write it.  After all, you won’t be reading what you’re writing.  Instead, your writing will be a summary of what’s been said!

With the slower pace, which has a higher engagement because students are using the time to carefully take notes, comes better questions from students.  In response to these questions you can naturally annotate the notes throughout the lesson, highlighting the source of the confusion for the students continually.

What all of this means is that by using a document camera, a lot of the elements of quality teaching are naturally accessible.  The pace is naturally improved to match the needs of students, the dialogue is improved, the exploration of misunderstanding is seamlessly incorporated into the lesson itself, without feeling tangential to the learning.

And all of that, especially the exploration of misunderstanding, provides the teacher with opportunity to provide for students what technology cannot do.  It allows you to easily step into a role that you must carve out for yourself when using more advanced technology.  The most important function of the teacher is to entice interest in students, to discover their motivations and to teach them instead of cover material.

How many times has this happened:  You teach a lesson.  The lesson is organized, complete, you’re proud of how it is constructed and delivered.  The students seem okay.  But when they test, the results are horrible.

This is what happens when we focus too much on the material, too little on the students.  For me, anyway, the document camera really helps me to focus on the students.  This is especially true with low-achieving students.  They need more help, a slower pace, a more responsive teacher.  Low-achieving students are less adaptive, flexible, and have less inclination to explore and challenge their understanding independently.

I am not saying, of the document camera, “This will revolutionize education.”  The document camera, like all technology, is only as good as it is used.

What I have tried to show here is how the document camera naturally offers you opportunity to perform what cannot be scripted, what cannot be programmed into a computer, what need an expert on a video cannot fulfill.  Your role as a teacher is to teach students, not cover material.

Whatever technology you use will fail to be effective if it is not used in a way that furthers that connection between students and content.  If the technology only improves exposure to content, does not help students to engage with the content in a way that is challenging and builds conceptual understanding, then it, too, will be ineffective.

The take away is, there is not replacement for a good teacher.  Tools that are used to enhance what a good teacher provides for a student are great.  Tools that lose sight of what quality teaching is, ultimately, hamper the educational process and harm students.

# Teaching Square Roots Conceptually

teaching square roots

How to Teach Square Roots Conceptually

If you have taught for any length of time, you’ll surely have seen one of these two things below.

Sure, this can be corrected procedurally.  But, over time, they’ll forget the procedure and revert back to following whatever misconception they possess that has them make these mistakes in the first place.

I’d like to share with you a few approaches that can help.   Keep in mind, there is no way to have students seamlessly integrate new information with their existing body of knowledge.  There will always be confusion and misunderstanding.  By focusing in on the very nature of the issues here, and that is lack of conceptual understanding and lack of mathematical literacy, we can make things smoother, quicker, and improve retention.

Step one is to teach students to properly read square roots.  Sure, a square root can be an operation, but it is also the best way to write a lot of irrational numbers.  Make sure you students understand these two ways of reading a square root number.

Students are quick studies when it comes to getting out of responsibility and side-stepping expectations.  Very quickly, when asked “What does the square root of 11 ask?” students will say, “What squared is the radicand?”

When pressed on the radicand, they may or may not understand it is 11.  But, they’ll be unlikely to have really considered the question for what it asks.  Do not be satisfied with students that are just repeating what they’ve heard.  Make them demonstrate what they know.  A good way to do so is by asking a question like the one below.

Another way to test their knowledge is to ask them to evaluate the following:

$\sqrt{2}×\sqrt{2}.$

We do not want students saying it is the square root of four at this point.  To do so means they have not made sense of the second fact listed about the number.  An alternative to using a Natural Number as the radicand is to use an unknown.  For example:

$\sqrt{m}×\sqrt{m}.$

Step two requires them to understand why the square root of nine, for example, is three.  The reason why it is true has nothing to do with steps.  Instead, the square root of nine asks, “What squared is 9?”  The answer is three.  There is no other reason.

Once again, students make excellent pull-toy dolls, saying random things when prompted.  Once in a while they recite the correct phrase, even though they don’t understand it, and we get fooled.  It is imperative to be creative and access their knowledge in a new way.

Before I show you how that can be done with something like the square root of a square number, let’s consider the objections of students here.  Students will complain that we’re making it complicated, or that we are confusing them.

First, we’re not making the math complicated.  Anything being learned for the first time is complicated.  Things only become simple with the development of expertise.  How complicated is it to teach a small child to tie their shoes?  But once the skill is mastered, it is done without thought.

The second point is that we are not confusing them, they are already confused.  They just don’t know it yet.   They will not move from being ignorant to knowledgeable without first working through the confusion.  If we want them to understand so they can develop related, more advanced skills, and we want them to retain what they’re learning, they have to understand.  They must grasp the concept.

So how can we really determine if they know why the square root of twenty-five is really five?  We do so by asking the same question in a new way.

Another way to get at the knowledge is by asking why the square root of 25 is not 6.  Students will say, “Because it’s five.”  While they’re right, that does not explain why the square root of 25 is not six.  Only when they demonstrate that 62 = 36, not 25, will they have shown their correct thinking.  But, as is the case with the other questions, students will soon learn to mimic this response while not possessing the knowledge.  So, you have to be clever and on your toes.  This point is worth laboring!

Step three involves verifying square root simplification of non-perfect squares.  This uncovers a slew of misconceptions, which will address. Before we get into that, here is exactly what I mean.

Have students explain what is true about the square root of twenty-four.  There are two ways they should be able to think of this number (and one of them is not as an operation, yet).

1.      What squared is 24?

2.      This number squared is 24.

The statement is true if “two times the square root of six, squared, is twenty-four.”  Just like the square root of 9 is three only because 32 = 9.

The first hurdle here is that students do not really understand irrational numbers like the square root of six.  They’ve learned how to approximate and do calculation with the approximations. Here is how they see it.

$\sqrt{2}=1.4$

$3+\sqrt{2}=4.4$

$3×\sqrt{2}=4.2$

What this means is that students believe:

1.      Addition of a rational number and an irrational number is rational.

2.      The product of a rational and irrational number is also rational.

a.       This can be true if the rational number is zero.

This misunderstanding, which naturally occurs as a byproduct of learning to approximate without understanding what approximation means, is a major hurdle for students.  It must be addressed at this time.

To do so, students need to be made to understand that irrational numbers cannot be written with our decimal or fraction system.  We use special symbols in the place of the number itself, because we quite literally have no other way to write the number.

A good place to start is with π.  This number is the ratio of a circle’s diameter and its circumference.  The number cannot be written as a decimal.  It is not 3.14, 22/7, or anything we can write with a decimal or as a fraction.  The square root of two is similar.  The picture below shows probably over 1,000 decimal places, but it is not complete.  This is only close, but not it.

Students will know the Pythagorean Theorem.  It is a good idea to show them how an isosceles right triangle, with side lengths of one, will have a hypotenuse of the square root of two.  So while we cannot write the number, we can draw it!

The other piece of new information here is how square roots can be irrational.  If the radicand is not a perfect square, the number is irrational.  At this point, we cannot pursue this too far because we’ll lose sight of our goal, which is to get them to understand irrational and rational arithmetic.

This point, and all others, will be novel concepts.  You will need to circle back and revisit each of them periodically.  Students only will latch on to correct understanding when they fully realize that their previously held believes are incorrect.  What typically happens is they pervert new information to fit what they already believed, creating new misconceptions.  So be patient, light-hearted and consistent.

Once students see that the square root of two is irrational, they can see how they cannot carry out and write with our number system, either of these two arithmetic operations:

This will likely be the first time they will understand one of the standards for the Number Unit in High School level mathematics.

Students must demonstrate that the product of a non-zero rational and irrational number is irrational.

Students must demonstrate the sum of a rational and an irrational number is irrational.

Keep in mind, this may seem like a huge investment of time at this point, and they don’t even know how to simplify a square root number yet.  However, we have uncovered many misconceptions and taught them what the math really means!  This will pay off as we move forward.  It will also help establish an expectation and introduce a new way to learn.  Math, eventually, will not be thought of as steps, but instead consequences of ideas and facts.

Back to our question:

Just like the square root of nine being three because 32 = 9, this is true if:

${\left(2\sqrt{6}\right)}^{2}=24.$

Make sure students understand that there is an unwritten operation at play between the two and the irrational number.  We don’t write the multiplication, which is confusing because 26 is just considered differently.  It isn’t 12 at all (2 times 6)!

Once that is established, because of the commutative property of multiplication,

$2\sqrt{6}×2\sqrt{6}=2×2×\sqrt{6}×\sqrt{6}.$

There should be no talk of cancelling.  The property of the square root of six is that if you square it, you get six.  That’s the first thing they learned about square root numbers.

$2×2×\sqrt{6}×\sqrt{6}=4×6.$

As mentioned before, students are quick studies.  They learn to mimic and get right answers without developing understanding. This may seem like a superficial and easy task, but do not allow them to trick themselves or you regarding their understanding.

A good type of question to ask is:

To do this, we students to square the expression on the left of the equal sign to verify it equals the radicand.  This addresses the very meaning of square root numbers.

Last step is to teach them what the word simplify means in the context of square roots.  It means to rewrite the number so that the radicand does not contain a perfect square.

The way to coach students to do this is to factor the radicand to find the largest square number.  This is aligned with the meaning of square roots because square roots ask about square numbers.  When they find the LARGEST perfect square that is a factor of the radicand, the rewrite the expression as a product and then simply answer the question asked by the square roots.  Here’s what it looks like.

$48=2×24,\text{\hspace{0.17em}}3×16,\text{\hspace{0.17em}}4×12,\text{\hspace{0.17em}}6×8.$

$\sqrt{48}=\sqrt{16}×\sqrt{3}$
Write the square root of the perfect square first so that you do not end up with
$\sqrt{3}4,$ which looks like $\sqrt{34}.$

$\sqrt{48}=4×\sqrt{3}$.

At this point, students should be ready to simplify square roots.  However, be warned about a common misconception developed at this point.  They’ll easily run the two procedures into one.  They often write things like:

$\sqrt{18}=\sqrt{9}×\sqrt{2}$

$\sqrt{18}=3\sqrt{2}$

${\left(3\sqrt{2}\right)}^{2}=9×2$

$9×2=18$

$\sqrt{18}=18.$

The moral of the story here is that to teach students conceptually means that you must be devoted, diligent and consistent with reverting back to the foundational facts, #1 and #2 at the beginning of this discussion.

This approach in no way promises to prevent silly mistakes or misconceptions.  But what it does do is create a common understanding that can be used to easily explain why $\sqrt{12}$ is not $3\sqrt{2}.$  It is not “three root two,” because

This referring to the conceptual facts and understanding is powerful for students. Over time they will start referring to what they know to be true for validation instead of examination of steps.  There is not a step in getting $\sqrt{12}=3\sqrt{2},$ that is wrong.  What is wrong is that their work is not mathematically consistent and their answer does not answer the question, what squared is twelve?

If a student really understands square roots, how to multiply them with other roots, and how arithmetic works irrational and rational numbers, the topics that follow go much more quickly.  After this will be square root arithmetic, like $5\sqrt{2}-3\sqrt{8},$ and then cube roots and the like.  Each topic that you can use to dig deep into the mathematical meaning will, over time, quicken the pace of the class.

In summary:

1.      Square roots have a meaning.  The meaning can be considered a question or a statement, and both need to be understood by students.

a.       This meaning is why the square root of 16 is 4.

2.      Square roots of non-square numbers are irrational.  Arithmetic with rational and irrational numbers is irrational (except with zero).

3.      To simplify a square root is to rewrite any factor of the radicand that is a perfect square.

a.       When rewriting, place the square root of the square number first.

4.      The simplification of a square root number is only right if that number squared is the radicand.

I hope you find this informative, thought-provoking, and are encouraged to take up the challenge of teaching conceptually!  It is well worth the initial struggles.

For lessons, assignments, and further exploration with this topic, please visit: https://thebeardedmathman.com/squareroots/

## This

Let’s talk about THE question in a teacher’s life … the baleful, “When am I going to use this in my real life?”

Yeah, that one.

The honest answer is probably, never… and they know it.  Why else ask, if not to subvert and diminish your role and purpose?  They don’t really want an answer.  What they want ... is just to watch you squirm, or to hear what B.S. you might spit out.  Either way, your class is now off the tracks!

However, it is a valid question, in its essence.  What is our purpose here?  It is a fair question, and one that needs to be answered.  And don’t be one of those people that thinks all content is applicable to daily life … it is NOT, nor should it be.  The purpose of education is not to train kids for every possible situation in life, but to equip them with an education so they can adapt to any possible situation.  The purpose of education is the development of the mind.  Sometimes we learn tricky things for the same reason some accountants lift heavy weights at the gym.

So let’s talk about how to change the answer to THE question.  Let’s turn it on its head.  Let’s answer the question: When am I going to use this in my real life, it in a way that swings the pendulum back in your direction.  Let’s answer the question in a way that stops the division and animosity that fosters the question in the first place.

See, the reality is, when a kid asks that question, they’re doing you a favor.  They’re providing you with insight that perhaps you’re serving the wrong this.  And if you’re not, whatever purpose you serve is not apparent to that student!

Let’s back up and take a larger scope view of the situation.  Especially in today’s educational climate, teachers stay in the classroom to be of service to students.  That’s it.  Teachers are blamed for all of the woes of society, for the failings of education, called lazy, and everybody seems to know what they’d do to fix it, if they were a teacher!

That sentiment, why teachers stay in the classroom, is the gateway to changing the answer to THE question.  Teachers are only in the classroom for the benefit of the students.  And surely, a teacher wouldn’t take on the sacrifices they do to stay teaching only to waste the time of their students, right?

Well … no, but kind of yes, too.

Let’s talk about job pressures … failing percentages in your classes, average scores on “high stakes tests.”   Those are big deals!  The test results are used to evaluate schools and teachers.  Administrators can be rewarded or fired on the basis of such things!

After all, good test results must be the sign of a great school.  Bad test results, well, that is really the teacher’s fault!  Yet, if a teacher holds a standard aligned with test results, the class failing rate will be too high, unacceptable, and a sign of bad teaching.  There will be unpleasant parent meetings; counselors, and administrators asking what’s being done to help the student, as if the student is hapless, a victim of the inevitable.

What am I doing to help this student?  I don’t know … showing up to work every day with lesson plans, a warm welcome and words of encouragement?  Oh, and I tell him to pick up his pencil fourteen times an hour, that’s a start, right?

Through either threat or blarney, bean-counters and pencil-pushers outside the classroom press hard to ensure that the teacher is performing due diligence to achieve high test grades.  Parents hover, students object, and through it all teachers are led to one inevitable observation: this is not why I teach.

Do those test scores really matter for students?  Sure, maybe ACT or SAT, AP, IGCSE or IB tests matter.  But those are the culmination of years of work.  Does it matter, to the student, if they pass their local state’s Common Core exam?  Not one bit.

In my real life, everybody will be impressed that I got a 3 on my Common Core State Test in English.

The purpose of an education is not to be able to pass a test.  The purpose of education is the development of the mind.  An educated person should be adaptable, thoughtful, able to communicate and appreciate various points of view other than their own.  An educated person should have perseverance and confidence, creativity and curiosity about the world around them.

A person that is educated should have an enriched life as a byproduct of their education.

When am I going to use this in my real life?

Well, that depends on what you mean by this.

Are you, as an educator, teaching this to help kids pass a test, or get a good grade in your class?  Are you teaching this to help them to know how to do their homework?  If so, there’s no judgement or blame.  Sometimes you have to make concessions just to get through the day.  We want kids to be successful.  The problem is, what are we using as markers of success?

But what if you could make this align with why you come back every year?

If you teach students about factoring polynomials so they can complete a worksheet, and maybe pass a quiz, your this is not powerful.  That is not why you teach.  Why make it what you teach?

The trick is to devise ways to teach kids how to think, to encourage creative problem solving and develop communication … to give them an education, while they learn how to factor a polynomial.

Personally, I never want a student to say to me, “Mr. Brown, you’re the only reason I got through math class.”  That’s too low of a standard.  That is exactly the this that makes THE question so damning to our efforts.  I don’t show up every day so my students can pass a quiz or test, or get a minimal passing grade in math class.

I show up every day to provide a challenge to my students, so they can test themselves and be better tomorrow than they were today.  And by better, I don’t mean greater proficiency at rationalizing the denominator. I mean of better mind.

To me, the best compliment a student can give a teacher is, “You taught me how to learn.”  In learning to learn, all of the pieces of an education are there.  To know how to learn you must be a problem solver, a critical thinker, be reflective, confident, and resourceful. A student that knows how to learn is prepared for an unknown future.

Ask yourself:  By teaching this, what’s being learned?  Are you just rehashing the same old lessons, just giving the same information the students could get on Khan Academy?  Are you asking them questions that can easily be answered by PhotoMath?  Are you printing another worksheet off of KutaSoft?

Challenge yourself to raise the bar.  Forget the bean-counters.  They’ll be happy when they see the results because when a student that knows how to learn takes a silly test, they do well!  Not only that, they’ll stand out when compared with students who were taught the content of the course only.

Unfortunately, if you’ve been dragged to the point where you realize, this is not why I got into teaching, and it consumes your day, you cannot answer THE student’s question honestly without using the word never.

You’re unlikely to find much guidance in the education industry that will change the this in THE question.  The industry sells books and professional development designed to get students to pass the test.  Their livelihood is generated from keeping the this we don’t want in THE question.

It is on us, educators dealing with students daily, to change the this.

The next time a student asks THE question … ask yourself, why?  Why did they ask?  Which this am I serving, the one designed for test scores, or the one educating students?

## Try to Solve This Problem … without Algebra

Can you solve the following, without doing any Algebraic manipulation?  Just by reading and thinking about what it says, can you figure out what x is?  (The numbers a, x, andare not zero.)

Given:  3ak

And:  ax = 4k

What must x be?

If you’re versed much at all in basic Algebra you will be tempted to substitute and solve.  After all, this is a system of equations.  But that will bypass the purpose and benefit of the exercise.

The intended benefit of this problem is that it promotes mathematical literacy, in particular, seeing relationships between terms.  It’s not a complicated relationship but it is of utmost importance to this problem.  Once you read and make sense of what the mathematical relationships are you can talk your way through the problem.

Once again, I believe the purpose of homework is learning.  Sure, sometimes it is practice and familiarity, but those are the only times that answer-getting is important.  Without understanding, having the right answer is often of little to no use.  If it were, then copying the answers from the back of the book would be sufficient for learning, right?

If you’re ready to see the solution, you can watch the video or read the text after the video.

I understand that sometimes it’s appropriate to read, but not listen or watch a video.  So here’s how this works.

Given:  3ak

And:  ax = 4k

The first statement says that the number k is three times bigger than the number a.  We don’t know what or k are but we know how they’re related and can think of lots of numbers that fit this relationship.  One number that’s three times as large as the other.

The number k is three times as big as the number a.

Think of this relationship one more way, for a moment.  The number k has two factors, 3 and a.  Whether is composite or prime is irrelevant really, it won’t change the fact that we could write k as the product of two numbers.  I mention this, not because it helps solve this problem but because it might.  Without knowing the path, sometimes it is a good idea to brain storm for a  moment and list as many things you know about the information given, before seeking an answer.  Sometimes, doing so, makes the answer apparent to you!

Let’s look at the second statement now.

Another number times a is four times as big as k.  This is perhaps a bit distracting, but the key information is there.  Remember, k is three times as big as a.  Now we have something four times larger than k.

Let’s look at this a different way.  The number 4k is not k at all, but instead, k and 4 are factors of new number.

If this new number is four times larger than k, and k is three times larger than a, how much larger is this new number than a?

You have three times as much money as me.  Bobert has four times as much as you do.  How much more money does Bob have than me?

For every dollar I have you have three.  For every dollar you have, Bobert has four.

Still don’t see it?  I know…picture good, word bad.  Here you go.

If 3ak, and ax = 4k, then is 12 because

## Exponents Part 2

two

Exponents
Part 2

Division

In the previous section
we learned that exponents are repeated multiplication, which on its own is not
tricky. What makes exponents tricky is
determining what is a base and what is not for a given exponent. It is imperative that you really understand
the material from the previous section before tackling what’s next. If you
did not attempt the practice problems, you need to. Also watch the video that review them.

In this section we are
going to see why anything to the power of zero is one and how to handle
negative exponents, and why they mean division.

What Happens with Division and Exponents?

Consider the following
expression, keeping in mind that the base is arbitrary, could be any number
(except zero, which will be explained soon).



3
5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaleqabaGaaGynaaaaaaa@37A0@

This equals three times itself five total times:



3
5

=33333

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaleqabaGaaGynaaaakiabg2da9iaaiodacqGHflY1caaIZaGaeyyX
ICTaaG4maiabgwSixlaaiodacqGHflY1caaIZaaaaa@4589@

Now let’s divide this by 3. Note that 3 is just 31.



3
5

3
1

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGymaaaaaaaaaa@395E@

If we write this out to seek a pattern that we can
use for a short-cut, we see the following:



3
5

3
1

=

33333

3

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGymaaaaaaGccqGH9aqpdaWcaaqaaiaaiodacqGHflY1caaIZaGaey
yXICTaaG4maiabgwSixlaaiodacqGHflY1caaIZaaabaGaaG4maaaa
aaa@4814@

If you recall how we explored reducing Algebraic
Fractions, the order of division and multiplication can be rearranged, provided
the division is written as multiplication of the reciprocal. That is how division is written here.



3
5

3
1

=
3
3

3333

1

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGymaaaaaaGccqGH9aqpdaWcaaqaaiaaiodaaeaacaaIZaaaaiabgw
SixpaalaaabaGaaG4maiabgwSixlaaiodacqGHflY1caaIZaGaeyyX
ICTaaG4maaqaaiaaigdaaaaaaa@48DF@

And of course 3/3 is 1, so this reduces to:



3
5

3
1

=3333=
3
4

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGymaaaaaaGccqGH9aqpcaaIZaGaeyyXICTaaG4maiabgwSixlaaio
dacqGHflY1caaIZaGaeyypa0JaaG4mamaaCaaaleqabaGaaGinaaaa
aaa@46EE@

The short-cut is:



3
5

3
1

=
3

51

=
3
4

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGymaaaaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaI1aGaeyOeI0

That is, if the bases
are the same you can reduce. Reducing
eliminates one of the bases that is being multiplied by itself from both the
numerator and the denominator. A general
form of the third short-cut is here:

Short-Cut 3:



a
m

a
n

=
a

mn

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGHbWaaWbaaSqabeaacaWGTbaaaaGcbaGaamyyamaaCaaaleqabaGa
amOBaaaaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaeyOeI0
IaamOBaaaaaaa@3F10@

This might seem like a
worthless observation, but this will help articulate the very issue that is
going to cause trouble with exponents and division.



3
5

3
1

=
3
5

÷
3
1

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGymaaaaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaI1aaaaOGaey
49aGRaaG4mamaaCaaaleqabaGaaGymaaaaaaa@4002@

.

But that is different than



3
1

÷
3
5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaaaaa@3B8A@

The expression above is the same as



3
1

3
5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaWaaWbaaSqabeaacaaIXaaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGynaaaaaaaaaa@395F@

This comes into play
because



3
1

3
5

=
3

15

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaWaaWbaaSqabeaacaaIXaaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGynaaaaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaIXaGaeyOeI0
IaaGynaaaaaaa@3DC0@

,

and 1



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaaaaaaaWdbiaa=nbiaaa@37C3@

5 = -4.

Negative Exponents?

In one sense, negative
means opposite. Exponents mean
multiplication, so a negative exponent is repeated division. This is absolutely true, but sometimes
difficult to write out. Division is not
as easy to write as multiplication.

Consider that 3-4
is 1 divided by 3, four times. 1 ÷ 3 ÷ 3
÷ 3 ÷ 3. But if we rewrite each of those
÷ 3 as multiplication by the reciprocal (1/3), it’s must cleaner and what
happens with a negative exponent is easier to see.



1÷3÷3÷3÷31
1
3

1
3

1
3

1
3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
pa4kaaiodacqGH3daUcaaIZaGaey49aGRaaG4maiabgEpa4kaaioda
cqGHsgIRcaaIXaGaeyyXIC9aaSaaaeaacaaIXaaabaGaaG4maaaacq
GHflY1daWcaaqaaiaaigdaaeaacaaIZaaaaiabgwSixpaalaaabaGa
aGymaaqaaiaaiodaaaGaeyyXIC9aaSaaaeaacaaIXaaabaGaaG4maa
aaaaa@5482@

This is classically repeated multiplication. While one times itself any number of times is
still one, let’s go ahead and write it out this time.



1
1
3

1
3

1
3

1
3

1

(

1
3

)

4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
SixpaalaaabaGaaGymaaqaaiaaiodaaaGaeyyXIC9aaSaaaeaacaaI
XaaabaGaaG4maaaacqGHflY1daWcaaqaaiaaigdaaeaacaaIZaaaai
abgwSixpaalaaabaGaaGymaaqaaiaaiodaaaGaeyOKH4QaaGymaiab
gwSixpaabmaabaWaaSaaaeaacaaIXaaabaGaaG4maaaaaiaawIcaca

This could also be written:



1
1
3

1
3

1
3

1
3

1

1
4

3
4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
SixpaalaaabaGaaGymaaqaaiaaiodaaaGaeyyXIC9aaSaaaeaacaaI
XaaabaGaaG4maaaacqGHflY1daWcaaqaaiaaigdaaeaacaaIZaaaai
abgwSixpaalaaabaGaaGymaaqaaiaaiodaaaGaeyOKH4QaaGymaiab
gwSixpaalaaabaGaaGymamaaCaaaleqabaGaaGinaaaaaOqaaiaaio

The second expression
is easier, but both are shown here to make sure you see they are the same.

Since 1 times 14
is just one, we can simplify this further to:



1

1
4

3
4

=
1

3
4

.

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
SixpaalaaabaGaaGymamaaCaaaleqabaGaaGinaaaaaOqaaiaaioda
daahaaWcbeqaaiaaisdaaaaaaOGaeyypa0ZaaSaaaeaacaaIXaaaba
GaaG4mamaaCaaaleqabaGaaGinaaaaaaGccaGGUaaaaa@40A3@

Negative exponents are
repeated division. Since division is hard to write and manipulate, we will
write negative exponents as multiplication of the reciprocal. In fact, if instructions say to simplify, you
cannot have a negative exponent in your final answer. You must rewrite it as multiplication of the reciprocal. Sometimes that can get ugly. Consider the following:



b

a

5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

To keep this clean, let us consider separating this
single fraction as the product of two rational expressions.



b

a

5

=
b
1

1

a

5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGIbaabaGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaGccqGH

The b is
not a problem here, but the other rational expression is problematic. We need to multiply by the reciprocal of



1

a

5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIXaaabaGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaaaaa@3981@

, which is just a5.



b

a

5

=
b
1

a
5

1

=
a
5

b

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGIbaabaGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaGccqGH
yyamaaCaaaleqabaGaaGynaaaaaOqaaiaaigdaaaGaeyypa0Jaamyy

.

This can also be
considered a complex fraction, the likes of which we will see very soon. Let’s
see how that works.



b

a

5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGIbaaqqaaaaaaOpGqSvxza8qabaGaamyyamaaCaaaleqabaGaeyOe
I0IaaGynaaaaaaaaaa@3C44@

Note:



a

5

=
1

a
5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
eB1vgapeGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaak8aacqGH
9aqpqqa6daaaaaGuLrgapiWaaSaaaeaacaaIXaaabaGaamyyamaaCa
aaleqabaGaaGynaaaaaaaaaa@4134@

Substituting this we get:



b

1

a
5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
daahaaWcbeqaaiaaiwdaaaaaaaaaaaa@3BB5@

This is b
divided by 1/a5.



b÷
1

a
5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaaa@3BB6@

Let’s multiply by the reciprocal:



b
a
5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

Now we will rewrite it in alphabetical order (a good
habit, for sure).



a
5

b

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

Let us consider one
more example before we make our fourth short-cut. With this example we could actually apply our
second short-cut, but it will not offer much insight into how these exponents
work with division.

This is the trickiest
of all of the ways in which exponents are manipulated, so it is worth the extra
exploration.



2
x

2

y

5

z

2

2

x
y
3

z

5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaWcbeqaaiabgkHiTiaaiwdaaaGccaWG6baabaGaaGOmamaaCaaale
ZaaaaOGaamOEamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaaaaa@45E3@

As you see we have four
separate bases. In order to simplify
this expression we need one of each base (2, x, y, z), and all positive exponents. So let’s separate this into the product of
four rational expressions, then simplify each.



2
x

2

y

5

z

2

2

x
y
3

z

5

2

2

2

x

2

x

y

5

y
3

z

z

5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaWcbeqaaiabgkHiTiaaiwdaaaGccaWG6baabaGaaGOmamaaCaaale
ZaaaaOGaamOEamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaGccqGHsg
IRdaWcaaqaaiaaikdaaeaacaaIYaWaaWbaaSqabeaacqGHsislcaaI
YaaaaaaakiabgwSixpaalaaabaGaamiEamaaCaaaleqabaGaeyOeI0
aSqabeaacqGHsislcaaI1aaaaaGcbaGaamyEamaaCaaaleqabaGaaG
beaacqGHsislcaaI1aaaaaaaaaa@5ED4@

The base of two first:



2

2

2

2÷
2

2

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIYaaabaGaaGOmamaaCaaaleqabaGaeyOeI0IaaGOmaaaaaaGccqGH
sgIRcaaIYaGaey49aGRaaGOmamaaCaaaleqabaGaeyOeI0IaaGOmaa
aaaaa@40D5@

We wrote it as
division. What we will see is dividing
is multiplication by the reciprocal, and then the negative exponent is also
dividing, which is multiplication by the reciprocal. The reciprocal of the reciprocal is just the
original. But watch what happens with the
sign of the exponent.

First we will rewrite
the negative exponent as repeated division.



2÷
1

2
2

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaaa@3B5E@

Now we will rewrite
division as multiplication by the reciprocal.



2
2
2

=
2
3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aSqabeaacaaIZaaaaaaa@3D58@

Keep in mind, this is
the same as 23/1.

We will offer similar
treatment to the other bases.

Consider first



x

2

x

=

x

2

1

1
x

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WG4bWaaWbaaSqabeaacqGHsislcaaIYaaaaaGcbaGaamiEaaaacqGH

Negative exponents are division, so:



x

2

x

=

x

2

1

1
x

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WG4bWaaWbaaSqabeaacqGHsislcaaIYaaaaaGcbaGaamiEaaaacqGH

Notice the x that is already dividing (in the
denominator) does not change. It has a
positive exponent, which means it is already written as division.



x

2

1

1
x

1

x
2

1
x

=
1

x
3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WG4bWaaWbaaSqabeaacqGHsislcaaIYaaaaaGcbaGaaGymaaaacqGH
flY1daWcaaqaaiaaigdaaeaacaWG4baaaiabgkziUoaalaaabaGaaG
aeaacaaIXaaabaGaamiEaaaacqGH9aqpdaWcaaqaaiaaigdaaeaaca
WG4bWaaWbaaSqabeaacaaIZaaaaaaaaaa@4A23@

This is exactly how simplifying the y and z will operation.



2
3

1

1

x
2

x

1

y
5

y
3

z
z
5

1

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIYaWaaWbaaSqabeaacaaIZaaaaaGcbaGaaGymaaaacqGHflY1daWc
aaqaaiaaigdaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyyXIC
TaamiEaaaacqGHflY1daWcaaqaaiaaigdaaeaacaWG5bWaaWbaaSqa
beaacaaI1aaaaOGaeyyXICTaamyEamaaCaaaleqabaGaaG4maaaaaa
caaI1aaaaaGcbaGaaGymaaaaaaa@5256@

Putting it all together:



2
x

2

y

5

z

2

2

x
y
3

z

5

=

2
3

z
6

x
3

y
8

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaWcbeqaaiabgkHiTiaaiwdaaaGccaWG6baabaGaaGOmamaaCaaale
ZaaaaOGaamOEamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaGccqGH9a
aSqabeaacaaI2aaaaaGcbaGaamiEamaaCaaaleqabaGaaG4maaaaki

.

Short-Cut 4:
Negative exponents are division, so they need to be rewritten as multiplication
by writing the reciprocal and changing the sign of the exponent. The last common question is what happens to
the negative sign for the reciprocal?
What happens to the division sign here:



3÷5=3×
1
5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
pa4kaaiwdacqGH9aqpcaaIZaGaey41aq7aaSaaaeaacaaIXaaabaGa
aGynaaaaaaa@3F12@

.
When you rewrite division you are writing it as multiplication. Positive exponents are repeated
multiplication.



a

m

=
1

a
m

,
1

a

m

=
a
m

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaleqabaGaeyOeI0IaamyBaaaakiabg2da9maalaaabaGaaGymaaqa
UaaGPaVlaaykW7daWcaaqaaiaaigdaaeaacaWGHbWaaWbaaSqabeaa
gaaaaaaa@4A81@

This is the second to
last thing we need to learn about exponents.
However, a lot of practice is required to master them fully.

To see why anything to
the power of zero is one, let’s
consider:



3
5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaleqabaGaaGynaaaaaaa@37A0@

This equals three times itself five total times:



3
5

=3"#x22C5;3333

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaleqabaGaaGynaaaakiabg2da9iaaiodacqGHflY1caaIZaGaeyyX
ICTaaG4maiabgwSixlaaiodacqGHflY1caaIZaaaaa@4589@

Now let’s divide this by 35.



3
5

3
5

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGynaaaaaaaaaa@3962@

Without using short-cut 3, we have this:



3
5

3
5

=

33333

33333

=1

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGynaaaaaaGccqGH9aqpdaWcaaqaaiaaiodacqGHflY1caaIZaGaey
yXICTaaG4maiabgwSixlaaiodacqGHflY1caaIZaaabaGaaG4maiab
gwSixlaaiodacqGHflY1caaIZaGaeyyXICTaaG4maiabgwSixlaaio
daaaGaeyypa0JaaGymaaaa@55F5@

Using short-cut 3, we have this:



3
5

3
5

=
3

55

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGynaaaaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaI1aGaeyOeI0
IaaGynaaaaaaa@3DC7@

Five minutes five is zero:



3

55

=
3
0

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaleqabaGaaGynaiabgkHiTiaaiwdaaaGccqGH9aqpcaaIZaWaaWba
aSqabeaacaaIWaaaaaaa@3BFF@

Then 30 = 1.

Τhe
3 was an arbitrary base. This would work
with any number except zero. You cannot
divide by zero, it does not give us a number.

The beautiful thing
about this is that no matter how ugly the base is, if the exponent is zero, the
answer is just one. No need to simplify or perform calculation.



(

3

2x1

e

πi

n=1

1

n
2

)

0

=1

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
HiLdaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIWaaaaOGaeyyp
a0JaaGymaaaa@4DBA@

Let’s take a quick look
at all of our rules so far.

 Short-Cut Example  a m ⋅ a n = a m+n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamyBaaaakiabgwSixlaadggadaahaaWcbeqaaiaad6ga aaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaey4kaSIaamOBaa aaaaa@4140@  5 8 ⋅5= 5 8+1 = 5 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaCa aaleqabaGaaGioaaaakiabgwSixlaaiwdacqGH9aqpcaaI1aWaaWba aSqabeaacaaI4aGaey4kaSIaaGymaaaakiabg2da9iaaiwdadaahaa WcbeqaaiaaiMdaaaaaaa@41C8@  ( a m ) n = a mn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbWaaWbaaSqabeaacaWGTbaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaWGUbaaaOGaeyypa0JaamyyamaaCaaaleqabaGaamyBaiaad6 gaaaaaaa@3EB7@  ( 7 2 ) 5 = 7 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI3aWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaI1aaaaOGaeyypa0JaaG4namaaCaaaleqabaGaaGymaiaaic daaaaaaa@3D93@  a m a n = a m−n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGHbWaaWbaaSqabeaacaWGTbaaaaGcbaGaamyyamaaCaaaleqabaGa amOBaaaaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaeyOeI0 IaamOBaaaaaaa@3F11@  5 7 5 2 = 5 7−2 = 5 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aWaaWbaaSqabeaacaaI3aaaaaGcbaGaaGynamaaCaaaleqabaGa aGOmaaaaaaGccqGH9aqpcaaI1aWaaWbaaSqabeaacaaI3aGaeyOeI0 IaaGOmaaaakiabg2da9iaaiwdadaahaaWcbeqaaiaaiwdaaaaaaa@4087@  a −m = 1 a m  &   1 a −m = a m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaeyOeI0IaamyBaaaakiabg2da9maalaaabaGaaGymaaqa aiaadggadaahaaWcbeqaaiaad2gaaaaaaOGaaeiiaiaabAcacaqGGa GaaeiiamaalaaabaGaaGymaaqaaiaadggadaahaaWcbeqaaiabgkHi Tiaad2gaaaaaaOGaeyypa0JaamyyamaaCaaaleqabaGaamyBaaaaaa a@4637@  4 −3 = 1 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaeyOeI0IaaG4maaaakiabg2da9maalaaabaGaaGymaaqa aiaaisdadaahaaWcbeqaaiaaiodaaaaaaaaa@3C0F@  a 0 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGimaaaakiabg2da9iaaigdaaaa@398F@ 50 = 1

Let’s try some practice
problems.

Instructions: Simplify the following.

1.



(

2
8

)

1/3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIYaWaaWbaaSqabeaacaaI4aaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaaIXaGaai4laiaaiodaaaaaaa@3B8D@

2.



3
x
2

(

3
x
2

)

3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
4bWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabe
aacaaIZaaaaaaa@400E@

3.



5

5
m

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI1aaabaGaaGynamaaCaaaleqabaGaamyBaaaaaaaaaa@38A4@

4.



5
2

x

3

y
5

5

3

x

4

y

5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI1aWaaWbaaSqabeaacaaIYaaaaOGaamiEamaaCaaaleqabaGaeyOe
WaaWbaaSqabeaacqGHsislcaaIZaaaaOGaamiEamaaCaaaleqabaGa
aaaaaa@44E1@

5.



7÷7÷7÷7÷7÷7÷7÷7

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
pa4kaaiEdacqGH3daUcaaI3aGaey49aGRaaG4naiabgEpa4kaaiEda
cqGH3daUcaaI3aGaey49aGRaaG4naiabgEpa4kaaiEdaaaa@4B9C@

6.



9
x
2

y÷9
x
2

y

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
daahaaWcbeqaaiaaikdaaaGccaWG5baaaa@3F94@

7.



9
x
2

y÷(

9
x
2

y

)

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaaaa@411D@

8.



(

x
2

2
x
6

)

2

(

x
2

2
x
6

)

2

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
cqGHflY1caaIYaGaamiEamaaCaaaleqabaGaaGOnaaaaaOGaayjkai
aawMcaamaaCaaaleqabaGaeyOeI0IaaGOmaaaaaaa@4BF0@

9.



(

a
m

)

n

a
m

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGHbWaaWbaaSqabeaacaWGTbaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaWGUbaaaOGaeyyXICTaamyyamaaCaaaleqabaGaamyBaaaaaa
a@3F08@

10.



(

3
x
2

+4

)

2

(

3
x
2

+4

)

3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aGinaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqaamaabm
0aaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaaaaa@4390@

## Exponents Part 1

exponents part 1

Exponents Part 1

One of the biggest
things to understand about math is how it is written. The spatial arrangement of characters is syntax. Syntax, in English, refers to the arrangements
of words to convey meaning.

Exponents are just a
way of writing repeated multiplication.
If we are multiplying a number by itself repeatedly, we can use an
exponent to tell how many times the number is being multiplied. That’s it.
Nothing tricky exists with exponents, no new operations or concepts to
tackle. If you’re familiar with
multiplication and its properties, exponents should be accessible.

That said, it is not
without its pitfalls. A balance between conceptual
understanding and procedural short-cuts is needed to avoid those pitfalls. The only way to strike that balance is
through a careful progression of exercises and examples. An answer-getting mentality will lead to big
troubles with exponents. People wishing
to learn how exponents work must seek understanding.

Let’s establish some
facts that will come into play with this first part of exponents.

1.
Exponents are repeated multiplication

2.
Multiplication

3.
counting

To simplify simple
expressions with exponents you only need to know a few short-cuts, but to
recall and understand, we need more.
These facts are important.

With an exponential
expression we have a base, the number being multiplied by itself, and the
exponent, the small number on the top right of the base which describes how
many times the base is being multiplied by itself.



a
5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaleqabaGaaGynaaaaaaa@37C9@

The number a is the base. We don’t know what a is other than it is a number.
It’s not a big deal that we don’t know exactly what number it is, we

Five is the exponent, which
means there are five a’s, all
multiplying together, like this:



aaaaa.

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
caGGUaaaaa@444F@

Something to keep in
mind is that this expression equals another number. Since we don’t know what a is, we cannot find out exactly what it is, but we do know it’s a
perfect 5th power number, like 32.
See, 25 = 32.

What if we had another number
multiplying with a5, like
this:



a
5

b
3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaaaaa@3BEE@

If we write this out,
without the exponents we see we have 5 a’s
and 3 b’s, all multiplying together. We don’t know what a or b equals, but we do
know they’re multiplying so we could change the order of multiplication
(commutative property) or group them together in anyway we wish (associative
property) without changing the value.



a
5

b
3

=aaaaabbb

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
eyyXICTaamOyaaaa@5437@

And these would be the same:



(

aa

)
(

aaa

)
(

bbb

)

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGHbGaeyyXICTaamyyaaGaayjkaiaawMcaamaabmaabaGaamyyaiab



(

aa

)
[
(

aaa

)
(

bbb

)
]

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
wMcaaaGaay5waiaaw2faaaaa@4F29@



(

aa

)

[
ab
]

3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x



a
2

[
ab
]

3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

This is true because
the brackets group together the a and
b, making them both the base. The brackets put them together. The base is ab, and the exponent is 3. This
means we have ab multiplied by itself
three times.

Keep in mind, these are
the intended meaning behind the spatial arrangement of bases, parenthesis and
exponents.

Now, the bracketed expression
above is different than ab3,
which is



abbb

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

.



(

ab

)

3

a
b
3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGHbGaamOyaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4maaaakiab

Let’s expand these exponents and see why this is:



(

ab

)

3

a
b
3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGHbGaamOyaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4maaaakiab

Write out the base ab times itself three times:



(

ab

)
(

ab

)
(

ab

)
abbb

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaaa@4C39@

The commutative property of multiplication allows us
to rearrange the order in which we multiply the a’s and b’s.



aaabbbabbb

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGIbGaeyyXICTaamOyaaaa@5310@

Rewriting this repeated multiplication we get:



a
3

b
3

a
b
3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
GjsUcaWGHbGaamOyamaaCaaaleqabaGaaG4maaaaaaa@3E2A@

The following, though, is true:



(

a
b
3

)
=a
b
3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGHbGaamOyamaaCaaaleqabaGaaG4maaaaaOGaayjkaiaawMcaaiab

On the right, the a has only an exponent of 1. If you do not see an exponent written, it is
one. If we write it out we see:



(

abbb

)
=abbb

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
GaeyyXICTaamOyaaaa@4D78@

In summary of this
first exploration, the base can be tricky to see. Parenthesis group things together. An exponent written outside the parenthesis
creates all of the terms inside the parenthesis as the base. But if numbers are multiplying, but not
grouped, and one has an exponent, the exponent only belongs to the number just
below it on the left. For example,



4
x
3

,

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

the four has an exponent of just one, while
the x is being cubed.

Consider:



(

x+5

)

3

.

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WG4bGaey4kaSIaaGynaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4m
aaaakiaac6caaaa@3BC4@

This means the base is x + 5 and it is multiplied by itself three times.



(

x+5

)

3

=(

x+5

)
(

x+5

)
(

x+5

)

(

x+5

)

3

x
3

+
5
3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
caaIZaaaaOGaeyypa0ZaaeWaaeaacaWG4bGaey4kaSIaaGynaaGaay
jkaiaawMcaamaabmaabaGaamiEaiabgUcaRiaaiwdaaiaawIcacaGL
WaaeWaaeaacaWG4bGaey4kaSIaaGynaaGaayjkaiaawMcaamaaCaaa
GccqGHRaWkcaaI1aWaaWbaaSqabeaacaaIZaaaaOGaaeiiaiaabcca
caqGGaaaaaa@55E5@

Repeated Multiplication Allows Us Some Short-Cuts

Consider the expression:



a
3

×
a
2

.

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaaa@4153@

If we wrote this out,
we would have:



aaa×aa

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
0cbbOpaaaaaasvgza8GacaWGHbGaeyyXICTaamyyaaaa@483B@

.

(Note: In math we don’t use colors to differentiate
between two things. A red a and a blue a
are the same. These are colored to help
us keep of track of what’s happening with each part of the expression
.)

This is three a’s multiplying with another two a’s.
That means there are five a’s
multiplying.



a
3

×
a
2

=
a
5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
qpcaWGHbWaaWbaaSqabeaacaaI1aaaaaaa@4379@

Before we generalize
this to find the short-cut, let us see something similar, but is a potential
pitfall.



a
3

×
b
2

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

If we write this out we get:



aaa×bb

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
0cbbOpaaaaaasvgza8GacaWGIbGaeyyXICTaamOyaaaa@483D@

This would not be an
exponent of 5, in anyway. An exponent of
five means the base is being multiplied by itself five times. Here we have an a as a base, and three of those multiplying, and a b as a base, and two of those
multiplying. Not five of anything.

The common language is that
if the bases are the same we can add the exponents. This is a hand short-cut, but if you forget
where it comes from and why it is true, you’ll undoubtedly confuse it with some
of the other short-cuts that follow.

Short-Cut 1: If the bases are the same you can add the
exponents. This is true because
exponents are repeated multiplication and the associative property says that
the order in which you group things does not matter (when multiplying).



a
m

×
a
n

=
a

m+n

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaey4kaSIaamOBaa
aaaaa@410D@

The second short-cut
comes from groups and exponents.



(

a
3

)

2

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGHbWaaWbaaSqabeaacaaIZaaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaaIYaaaaaaa@3A43@

This means the base is a3, and it is being multiplied by itself.



a
3

×
a
3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaaaaa@3BB8@

Our previous short cut said that if the bases are
the same, we can add the exponents because we are just adding how many of the
base is being multiplied by itself.



a
3

×
a
3

=
a

3+3

=
a
6

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaaIZaGaey4kaSIaaG4maa

But this is not much of
a short cut. Let us look at the original
expression and the outcome and look for a pattern.



(

a
3

)

2

=
a
6

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGHbWaaWbaaSqabeaacaaIZaaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaaIYaaaaOGaeyypa0JaamyyamaaCaaaleqabaGaaGOnaaaaaa
a@3D26@

Short-Cut 2: A power raised to another is multiplied.



(

a
m

)

n

=
a

m×n

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGHbWaaWbaaSqabeaacaWGTbaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaWGUbaaaOGaeyypa0JaamyyamaaCaaaleqabaGaamyBaiabgE

Be careful here,
though:



a

(

b
3

c
2

)

5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

=



a
b

15

c

10

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aGymaiaaicdaaaaaaa@3BFF@

Practice Problems

 1.        x 4 ⋅ x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGinaaaakiabgwSixlaadIhadaahaaWcbeqaaiaaikda aaaaaa@3C18@ 8.  ( 5xy ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI1aGaamiEaiaadMhaaiaawIcacaGLPaaadaahaaWcbeqaaiaaioda aaaaaa@3B23@ 2.        y 9 ⋅y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa aaleqabaGaaGyoaaaakiabgwSixlaadMhaaaa@3B36@ 9.  ( 8 m 4 ) 2 ⋅ m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI4aGaamyBamaaCaaaleqabaGaaGinaaaaaOGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaakiabgwSixlaad2gadaahaaWcbeqaaiaaio daaaaaaa@3F41@ 3.        z 2 ⋅z⋅ z 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaCa aaleqabaGaaGOmaaaakiabgwSixlaadQhacqGHflY1caWG6bWaaWba aSqabeaacaaIZaaaaaaa@3F64@ 10.  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## The Purpose of Homework and My Response

The purpose of homework is to promote learning.  That’s it.  It’s not a way to earn a grade or something to keep kids busy.  It’s also not something that just must be completed in order to stay out of trouble.  Homework is a chance to try things independently, make mistakes and explore the nature of those mistakes in order to better learn the material at hand.

If students are not learning from the homework, it is a waste of time and effort.  There are a few things that could cause students not to learn from the homework.  Even if the assignments are of high quality, without the reflection and correction piece, students will not learn much from homework.

Reflection and correction go together.  It’s not about getting right answers, but thinking about what caused mistakes, identifying misconceptions or procedural inefficiencies and replacing those.  To reflect a student should NOT erase their incorrect working but instead should write on their homework, in pen, what went wrong and what would have been better.

It is quite possible more can be learned when reviewing homework than any other time.  It is certainly a powerful experience.

Textbooks and videos, tutors and peer help offer little appropriate support to help make homework, or practice, meaningful.  Textbooks only provide correct answers, YouTube videos usually do similar treatment to topics as textbooks offer.

I wish to help students learn and believe that reviewing work that has been done is too powerful of an opportunity to pass.  The trick is, how can I provide reflection and insight when to someone I am not sitting with and talking to?  I think I can help provide this reflection piece by doing all of the practice problems myself on a document camera and discussing pitfalls and mistakes, as well as sharing my thinking about the problems as I tackle them.  Further, I can share typical mistakes I see from students as they are learning topics.

So as I develop the Algebra 1 content I will be working on adding videos and short written responses to the assignments to help students think about what they’ve done, its appropriateness, correctness and their level of understanding.