The Toaster Problem in Education

It’s easy to talk about shifting education towards a more concept based approach.  But it’s hard to see what that really means in practice.  I’m not a betting man, but would be willing to bet that upon inspection there are many things you think you understand conceptually in your topic, but you just feel that way because you understand procedure well enough to always arrive at correct answers.

I can offer an example in math:  Why does the order of operations work?  Why does the structure in the order of operations guide us to the correct calculation?

Let’s use a mathematical way of thinking to approach this problem of understanding what conceptual approach education looks like, compared to our current procedure based approach.  (Tell-tale sign that you’re procedurally based is if your students cannot remember how to do something big a year later.  Or, do you consider the work assigned before your lesson?)

Imagine you want your students to know how to make toast.  You could introduce them to a toaster.  Then, demonstrate how the bread-item is dropped in the slots on the top, the little knobby is turned to select the desired level of darkness, the button with the picture of the type of bready-material being toasted is pushed, and the lever is depressed.

If it’s an advanced class, maybe some discussion is given to what to do if the toast gets stuck, and why you should always unplug the toaster when finished with it because toasters have notoriously cheap circuits that short out, causing a fire.

Oh, one last thing.  All toasters are good toasters.  There are no bad toasters.  Some make light toast, some dark toast.  If you show preference to one kind of toaster, you’re then the exception to the rule because we tolerate everything except intolerance.

That’s a very typical American style of teaching something.  We cover how to use a tool and throw in a little social justice message to boot.  (That is not a comment on the need for awareness of social issues except to say that math textbooks are inappropriate platforms for them.)

Imagine that instead of wanting your students to know how to make toast, you wanted them to know about toast.   You teach them what it is, previously cooked bread that is now slightly, but evenly, burned on the cut-faces making a slightly stiffer, crunchier piece of substrate for the delicious spreadable material of your choosing.

For the sake of this thought experiment, let’s say you also show them a toaster, but that’s it.

Now consider a pair of students.  One who learned the first method, and the second learned about toast, but spent little time with a toaster.

Which student could make toast if the toaster broke?  Which understands what a toaster really does?

Teaching how to use a toaster is procedural, while teaching what toast is would be conceptual.

Education is a HUGE industry with an enormous amount of inertia to overcome before change is realized.  There are jobs at stake if responses to changes go wrong.  Companies invest millions to supply the desires of schools.  And what do schools want?  They want to be like everybody else, because it’s safe!

We have these methods, that if not effective, are at least safe because we have used them for a long time, so has everybody else.  So if we’re close to the average, we’re okay.

But don’t get me wrong, things in education will change.  Pretty soon curriculum will be all conceptual.  Kids will be reinventing the wheel at every turn.  We see some of that in the elementary levels right now.  That’s truly a shame because it’s harmful.  Young kids do not yet possess the faculties for abstraction!  They need to know how to use a 3rd grade toaster, if you will.

I am NOT a doomsday preacher here, but I do not believe education cannot fix itself.  It is so established in the way it operates that the path we are on will remain until something really big from way up high changes.  The likelihood of that being a good change is slim because politicians aren’t educators.  Even if the idea is good, from above, the execution will be poor because it’s ideas, not how they play out, that gets people elected.

But, the change from teaching how to use a toaster to teaching what toast is, well, is needed.  Even for students to pass the new style of standardized testing they need to know what toast is.

Beyond that, for them to be successful in college, the nature of toast must be understood.  To change math from a hurdle to an opportunity, they’ve got to know all about toast, not just how to use the toaster.

It is these last two things, the belief that the education system cannot right itself, and the need for conceptual understand, that has motivated me to step outside of education for my project.

Why Good Lessons Fail

Ever had a lesson you were THRILLED about?  You loved it, it was fantastic, interesting, crisp, approachable and ... wonderful in every possible fashion.  And yet, when you delivered that lesson, it flopped!

What gives?  What was wrong with the lesson?

In reality, there was probably nothing wrong with the lesson.  Sure, all can be improved, but the lesson wasn't the problem, the delivery was.  It seems there exists an inverse relationship between how much I love a lesson and how well received it is.  The more I love it, the more students hate it!

What it really boils down to is engagement.  We are so sure that what we have to say will blow minds, that we forget our number one task ... making sure we are teaching students, not just covering material.  We assume that because we find it interesting and fascinating, and because we had such a grand time putting the lesson together that they'll gravitate towards it.

But gravitate towards it in favor of what?  What captures the attention of students?  Drama at lunch, fights with family members, changes in weather, they might be tired from staying up and watching the new season of Stranger Things on Netflix ... we don't know.  But whatever has their attention, we must wrestle it away.

In a normal lesson we are usually vigilant and on top of distractions and such.  We work hard to make the lesson itself interesting.  But in a lesson that needs no such adornments, we fail to sell it.

So regardless of whether you think it's great, they need to be sold on the fact!

There are a couple things that you can do, at any point in time, if they're not engaged.  These work for average and poor lessons, not just the great ones that we hope will inspire a future generation of (whatever it is you teach).

Before I share with you three ways to quickly grab their attention, let me say that once you have it, you can just jump right back into the lesson.  You'll have their attention, they'll not even notice that suddenly they're learning stuff!

My favorite, go-to, method of grabbing attention is with a quick, cheesy, usually Dad-Joke.  I sometimes look up a bunch of them, print them off and have them at the ready.  There are a few that I have on the ready at any given moment, but since I don't often tell them outside of the classroom, I forget.

Make it short and dumb, they'll be captured, even if they complain.  Then, back to the lesson.

And with all of these, you just jump right into the attention getting performance, you can do it mid-sentence if you please.

The second method is with a quick story about something interesting.  It can be that you wanted some cereal for breakfast and there was only a splash of milk left in the fridge!  So you couldn't even have dry cereal, just slightly less than soggy junk -- How FRUSTRATING!?!?!  Get some feedback and jump right back in.

The last method I use is direct.  I simply tell them they're distracted and that they need to do their best to focus.  I'll sell why (perhaps the material is dry but will be very important and interesting in context later, or some other reason).  I'll share that I feel the same way, burned out and tired, but explain that we all have a job to do.  "Let's just get through these next few parts and we're done for the day, if we do them well.  If not, we'll have to revisit this again in the near future."

Whatever methods you use, mix it up.  If you become too predictable with these they'll fail to gather attention.  So, "Stay frosty," like the line in Aliens suggests.

Anyhow, I hope these are helpful tips.  Just remember, no how great your lesson is, engagement is still the most important part of the lesson.  Without it, they'll not learn anything!

What is Algebra?

This past month has been very busy here for The Bearded Math Man.  I’ve learned a lot about things I have merely taken for granted and have shared most of them with you here on my site.  And while I have a goal and a mission, the methods of achieving that goal are still forming.  I’m learning what works best and what doesn’t work.  One such thing I’ve discovered is the purpose of this blog.

This blog is meant for two audiences.  Those interested in math and those teaching math.  Now that I have that defined, I’ll keep a more focused range of topics.  I just thought that was worth mentioning.

Now, for today’s topic, Algebra.  I do not intend to teach you Algebra, but would like to share something I did not know about the subject.  Algebra means to make complete, or to resolve.  I knew it was named after a Persian mathematician in the early 9th century, but that the branch of mathematics goes farther back in time than the name itself, even the Babylonians used Algebriac concepts.  But I thought the name was just that, a name.

It is stunningly powerful to recognize what Algebra means.  Everything operation we perform in Algebra is to meet this end, to complete or resolve an equation!  That’s what we do when we’re solving an Algebraic Equation.

One other thing you may not have known about Algebra is the equal sign.  The symbol itself never appeared until the 16th century and it traveled the entire width of the page.  It is hard to imagine how this would be a more efficient way of describing the equality present between two things, but it was.  Over time it was shortened to what we have today.  This is more than just an interesting factoid, too.  It goes to show that sometimes great ideas are so revolutionary that they seem obvious in hindsight.  First, we have a symbol that means equals, then we have, over time, an easier way of writing that symbol.

In many ways, isn’t that what makes mathematics so difficult, the jargon and abstraction?  That’s why one of my main points of focus is instilling mathematical literacy in students.  If they can read the math for what it says, not just as a funky collection of shapes and symbols, the mathematical ideas present themselves in a sensible and approachable fashion.

That’s what I’ve tried to do with my introduction to Algebra as a branch of mathematics, which is taught in Algebra 1, the class.  Here’s the link to the page.

As always, I thank you for reading and hope I’ve stirred some curiosity in you.

PS:  If you are interested in some of the history behind Algebra, the following book is highly recommended.  If you purchase it through this link you will help support the mission here of changing math from a hurdle in the way of young peoples’ dreams to a platform upon which success is built, and at no additional cost to you.

Is Infinity Real?

How Many Primes are There
Is Infinity Real
Part 1

Teachers: The following is a discussion that can be had with students to create interest in mathematics by discussing two very easy to understand, but perplexing problems in mathematics.  First, the nature of infinity.  The second is the lack of pattern and order in the prime numbers.

The number of primes is infinite.  Euclid proved it in a beautiful, easily understood proof by contradiction.  Paraphrasing, he said that there are either infinitely many primes, or a finite number of primes.  So let’s pick one and explore it.  Say there are a finite number of prime numbers.  If you were to list them all, then take their product you would have a very large number.  But if you just add one to that number, it would be prime because none of the other prime numbers would be a factor of it.  It would have exactly two factors, one and itself.

In case you don’t believe this works, let’s say we can list all of the primes, but there are only four.  Let’s say the entire list of primes was 2, 3, 5, and 7.  Their product, 2 × 3 × 5 × 7 = 210.  This number is composite because all of the primes are factors of it.  Add one to it, arriving and 211 and none of the prime numbers are a factor of it…making it have the factors of 211 and 1.  That means it is prime.

So it is false that there are a finite number of primes. Therefore, the are infinitely many prime numbers.

Beautiful, right?  Case closed. … or is it?

The case is closed, if you believe infinity exists.  To be clear, infinity is not a number, it’s a concept.  A set can only approach infinity, nothing ever equals infinity because it’s an idea.  The idea behind infinity is that the collection of things just keeps growing and growing.

We, as humans, have a very big problem with very big numbers, even large groups of things.  For example, there are some things that we only have a plural word for, we do not possess a singular word for these things.  A few examples are rice, sand, hair, shrimp and fish.  You can have a single hair, a grain of sand (or rice), and so on.  They are so vast in quantity they become indistinguishable.

And yet, they’re finite. You could conceivably collect all of the sand in the world and count every grain.  More sand does not magically appear once it is all collected.

What about stars in the sky?  What we call the observable universe is how far we can see.  We don’t know if it goes on forever, or if it is somehow contained.  Perhaps the word, universe, is misleading.  Perhaps there are multiples of it, maybe as many as there are grains of sand on the earth.

Before we chase that rabbit down its hole, let’s get back to earth.  Euclid’s proof that there are infinitely many prime numbers is beautiful.  But is he right?  Surely his proof is flawless, but what about infinity.  We have no examples of infinity, it might just be a human construction.  Now, if mathematics can discover things that are real and applicable from such a thing, that’s all the more powerful the tool it is, but what if we’re wrong about infinity?  There are two things I want you to consider as we explore prime numbers and their relationship with infinity.

The first thing is:  There’s an axiom (a statement we just accept as truth), called the Axiom of Infinity.  It basically says that there are infinite sets of things, like natural numbers.  We just say it’s true and roll with it until we discover a problem.  Then, we either adjust our axiom or start a new one.

The second thing is:  In the early 20th century a man named Kurt Gödel showed that we cannot actually prove any system of mathematics is true without assuming some supporting evidence is true.  We have to assume something is true in order to know if other things are true, roughly speaking.  In order to know if the thing we assumed to be true is actually true or not (like infinity), we have to assume that something else, more basic, is true.  So, and I’m taking some liberties here to make my point, but a conclusion, like the number of primes being infinite, is only as worthy as the presupposition (infinities exist).

Let’s look at a few strings of prime numbers and see if we can’t get our heads around this whole infinity thing.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

The gaps between these prime numbers are below.

1, 2, 2, 4, 2, 4, 2, 4, 6

Another string would be:

907, 911, 919, 929, 937, 941, 947

The gaps here are listed below.

4, 8, 10, 8, 4, 6

They are still relatively close.  Many mathematicians have tried to find a pattern in prime numbers.  After all, if you can find a pattern, then you can find the next one.  How cool would that be, right?

You might be thinking, uh, why would that be cool?

Well, there’s big money being paid if you can find the next prime number.  There is a project called GIMPS (Great Internet Mersenne Prime Search), where you can participate in the search.  And if your computer finds the next prime, you get some cash!

The last prime found with GIMPS was in 2013.  (At the time of this being written, it is 2017.) The number is massive.  The text file of the digits in the number is 7.7 MB.  That’s more data that a song and this is just a list of numbers.  The number is 257,885,161 – 1.  The number is huge that to verify that it is prime takes massive super computers days upon days to perform the calculation.  Finding the next prime number is a huge undertaking, very complicated and difficult, requiring computers all over the world working together before one is discovered.

Why all the fuss? What good are they?

Well, they keep you from being robbed, for one.  Internet security uses prime numbers to encrypt (code) your banking information.  The merchant will have a huge number that they multiply your card number by (kind of).  The huge number is the product of two of these gigantic prime numbers.  It’s so big that even though everybody (would be thieves) know it’s the product of two primes, they can’t figure out which two numbers.  The encrypted number is sent to your financial institution, who knows which two primes were used, which is basically like a key.

It’s also weird, and cool, that some bugs have a life cycle that only occurs in prime numbers!  Cicadas only come out and breed, and then die, in prime number years.  Incredible.

Back on track, forgive me.  It feels there are infinitely many tangents I can follow with math!  We have not been able to find a pattern in the prime numbers yet and let’s take a look at why.  You see, as these primes get huge, the gaps get larger and larger…approaching infinity!

Let’s take a look at one more string of prime numbers.

10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099

 

The differences here are as follows.

28, 2, 22, 6, 10, 12, 2, 6

No discernable pattern, right?  If you can find one, you stand to make significant history, no one has found one yet.  We have some approximations that work within certain constraints but they all break down eventually.

But, to be clear, if you could find a pattern in the gaps between the primes a formula could be created that would generate prime numbers.  We can generate natural numbers by just adding one to the largest we have come up with so far.  But primes, as you’ve seen with the GIMPS project, aren’t so easily discovered.

And here’s one of the issues.  The gaps between prime numbers can get huge, perhaps infinitely huge.  Consider this.

Fact 1:  5! = 5×4×3×2 = 120

Fact 2:  120 is not prime because it is divisible by 5 and 4 and 3 and 2.

Fact 3:  5! + 5 is not prime because it is divisible by 5.  (When we add another 5, it’s like skip counting when you first learned multiplication.)

The same is true for 5! + 4 being divisible by 4, because 120/4 = 30.  5! + 4 is 4 × 31, there’s one more four.

The same holds true for 5! + 3 being divisible by 3 and 5! + 2 being divisible by 2.

Fact 4:  What all this means is that there after 5! + 1 there are four consecutive numbers that are composite.

This would also work for 100!  The number 100! + 100 would be composite.  For that matter, 100! + 37 would be composite also.  100! Plus all of the numbers up to and including 100 would be composite, (except possibly adding 1).

This means there is a gap of 99 after 100! + 1.

This goes on forever, arbitrarily large numbers, like 1,000,000,000,000!  There would be a gap of 1,000,000,000,000 – 1 numbers after this number that are composite.

We could write this in a general sense.  Let a and x be a whole numbers such that a is less than or equal to x.  (a x).

Then x! + a is composite.

Since x is a whole number and whole numbers are infinite, then there are infinitely large gaps between the large prime numbers, themselves being infinite.

Crazy, right?

So if the gaps between primes gets infinitely large, how can there be infinitely many prime numbers?

Well, there’s one more piece of information to be considered.  Twin primes are prime numbers that are just two numbers apart.  The primes 2 and 3 are only one apart, but all others are an even number apart, the smallest gap being a gap of two, like 5 and 7, or 11 and 13.

There’s a conjecture (not as strong as an axiom), that is yet unproven, but we’re getting closer, that states that there are an infinite number of twin primes.  The largest known pair of twin primes is below:

3,756,801,695,685 × 2666,689 – 1
and

3,756,801,695,685 × 2666,689 +1

Those numbers are too large to be written out!

While we do not yet know, with a proof, that there are infinitely many twin primes, we do know that there are infinitely many primes that have a maximum distance between them and it might be as low as a difference of sixteen.  This is all being discovered and explored and fought over at the moment.

So on one hand we have infinitely large gaps between prime numbers, but when they do pop up, they will do so in clumps and groups?

If all of this makes your head spin, then I have succeeded.  I am not trying to convince you that infinities do not exist, or that they do.  I am trying to show that math is contentious and changing.  As we learn and discover new things math is changing.  Math is just a language we use to describe the world around us.  So powerful is math that we are not even sure if it is a human invention at all or rather a discovery!

What are your thoughts?  Please share them in the comments below.

As always, thank you for your time. I hope this has stirred some thought, maybe even sparked a passion for mathematics!

 

At the time of the making of this video the world’s largest prime number is not the last one found by the GIMPS project.  However, they’re likely to find another even larger one, sometime soon.  There’s a video below (Largest prime number) that discusses that number and prints it out … it takes up as much paper as three large books!

For some fascinating and approachable treatment of prime numbers, consider the following videos:

Gaps between prime numbers: https://youtu.be/vkMXdShDdtY

The largest prime number:  https://www.youtube.com/watch?v=lEvXcTYqtKU

Infinite Primes:  https://www.youtube.com/watch?v=ctC33JAV4FI

Large Gaps Between Primes:  https://www.youtube.com/watch?v=BH1GMGDYndo

If you found this helpful and would like to help make these videos possible, to help break down the obstacle that math presents itself as to young people, please consider visiting my patreon site:

www.patreon.com/beardedmathman

The Problem with PEMDAS

The problem with PEMDAS

This problem has really stirred a lot of interest and created a buzz on the internet. I can see why, it’s an easy one to miss.  And yet, PEMDAS is such an easy thing to remember, the mnemonic devices offered make for a strong memory.  So people passionately defend their answers.

6 ÷ 2(2 + 1)

I am going to tell you the answer in just a moment, but before I do, please listen to why I think this is a worthy problem to explore.

There are two fundamental misconceptions with math that make math into a monster for so many people, and this problem touches on both.  In a sense, neither has anything to do with the order of operations specifically.

The first issue is understanding that spatial arrangements in math mean something.  The way we write the numbers and symbols has a meaning, very specific at that.  In this video by Mind Your Decisions, https://youtu.be/URcUvFIUIhQ, he shares where there was a moment in time when we used different conventions to write math.

And while math may or may not be a human invention, the symbols and arrangements and their meanings certainly are.  Just like the letter A is only a letter and with a specific sound because we all agree.  Just like a red light means stop, a green light means go and a yellow light means HURRY HURRY HURRY!

The second, and more over-arching issue here, is the misconception that addition and subtraction are different.  They are fundamentally the same thing.  Subtraction is really addition of opposite numbers.  Perhaps to shore this misconception negative numbers should be introduced instead of subtraction.

Now you might argue and say, Wait, addition has properties that subtraction lacks, like the commutative property.

You’re correct, 5 + 3 = 3 + 5, while 5 – 3 does not equal 3 – 5.  However, 5 – 3 is really five plus the opposite of three, like written below.

5 + - 3

And that is the same as this expression below.

-3 + 5

So the AS at the end of PEMDAS is really just A, or S, whichever leads to the better nursey rhyme type device to improve recall.

Since we believe that addition and subtraction are different, we also come away with the belief that multiplication and division are different.  Sorry, they’re not.  Division is multiplication of the reciprocal.  Remember that whole phrase from your school days? (How was that for a mnemonic device?)

And while division does not have the commutative property, that again is a consequence of the way we write math.  If we only wrote division as multiplication of the reciprocal, we would see that multiplication and division are in fact the same.

So, back to the problem.  The most common wrong answer is 1.  The correct answer is 9.  Here’s a great video on the order of operations, super catchy and articulates the importance of left to right as written for multiplication and addition.

Last thing:  Now, in creative writing the intent of the author must be considered, should it also be considered here?

Let me know what about this you like, dislike or disagree with.  Let me know what is helpful.  I really want to promote success through making math transparent.  It’s my mission.  You can help support my mission by just sharing and liking this.  Subscribe to my blog if you’re a teacher as I will be populating it with lots of teacher advice, not all math related.

Thank you again for reading.

The Square Root Club

If you’re a teacher, I have a short story that you can share, adapted to fit your own style, that you can use to address the biggest issue with teaching … students learn what they want to learn.  Creating interest in mathematics for teenagers can sometimes be a challenge.  One of the easiest ways to do so is with humor.  The following story is actually true, but humorous, and I think will create some curiosity and thus learning opportunities for students. 

I believe the appropriate audience would be pre-algebra students learning about square roots up to algebra students learning about square roots.  Anyhow, if you find this helpful, please let me know. 

The Square Root Club

My daughter, a senior at the University of Arizona, called and said she’d uncovered an issue in math that is both absolutely impossible and yet, true.  My interest piqued, I listened attentively as she asked if I’d ever heard of the square root club.

 

The square root club, I was informed, is a club of dubious membership.  To become a member the square root of your GPA must exceed your GPA.  What a delightful treat this was…and to think, I’d never heard of such a thing!

She continues to tell me that she met someone who was a member.  I asked her how she knew, because certainly her friend would be ignorant of his membership.  Surely, someone in the club would not be smart enough to be aware of the fact, right?

That’s what she said was the funniest part, the part that was seemingly impossible!  He knew about it, even made up the name of the club himself.  He was no longer a member, just graduated with his bachelor’s degree with a 3.0 GPA.

Note:  GPA (grade point average) is calculated by assigning a numerical value to letter grades.  An A is 4, B is 3, C is 2, D is 1 and an F is zero.

The moral of the story is that grades don’t reflect potential, they reflect what you show you know.  Many high school students get by with intelligence but never work.  Upon arriving in college they are overwhelmed, never having had to work hard or apply themselves.  Before they know it, they’re buried and there’s no quick fix like there can be in high school.

To that point, nobody cares about someone’s potential, not even your mother. Imagine your mom told you to clean your room.  Because she told you to do it, she believes you have the potential.  However, if you do not clean it, she will be satisfied by the fact that you could have cleaned it.

Now of course, the question being begged here is, what could his GPA have been?

How Math Fixed Music

How Math Fixed Music

Rational Exponents Sound GREAT

Before we dive in, music is primarily defined by what we hear, not by the analysis and insight provided by math.  For example, an octave is a note whose frequency is double that of its parent note.  The mathematical relationship was discovered after the fact.  The following is an exploration of how math is used in music, but I don’t want to put the cart before the horse here.  The math supports the music, makes it work.  But the math is really fine-tuning what we hear.

Pythagoras developed a musical system that over the years evolved into what we have today. (At least Pythagoras is often credited for it.)  Not until “recently,” however, has one of the major problems with music been resolved (see what I did there with resolve?).

The problem the ancients had is that their octaves didn’t line up.  An octave, as I mentioned early, is a note that has twice (or half) the frequency of another note.  Octaves, in modern western music, share the same names, too.  The note A, at 440 Hz, has an octave at 880 Hz, and also 220 Hz.   (There are infinitely many octaves, in theory, though our ears have a limited range of things we can hear.)  The ancients, however, had a problem because after a few octaves, well, they were no longer octaves.

In western music we have 12 semi-tones, A, A# (or B-flat), B, C, C#, D, D#, E, F, F#, G, G# and then A again.  It’s cyclic, repeated infinitely both higher and lower. Each semi-tone in the next series of 12 notes is an octave of our first series of notes.  And the relationship between notes is what makes them, well, musical, not just sounds.

The problem is defining that relationship.  You see, because each note is slightly higher (has a higher frequency), and each note’s octave is double that frequency, what happens is the notes get further and further apart (the differences in their frequencies increases).

Let’s take a look at the frequencies:

Note Frequency
A 220.00
 A# 233.08
B 246.94
C 261.63
 C# 277.18
D 293.66
 D# 311.13
E 329.63
F 349.23
 F# 369.99
G 392.00
 G# 415.30
A 440.00
 A# 466.16
B 493.88
C 523.25
 C# 554.37
D 587.33
 D# 622.25
E 659.25
F 698.46
 F# 739.99
G 783.99
 G# 830.61
A 880.00

As you can see, the differences between consecutive notes is increasing, at an increasing rate!  This is not a linear relationship.  Because of this, the ancients had a very hard time defining what was an A and what was a D, especially when you started moving around between octaves.  Things got jumbled, and out of tune.

It is tricky to find the proportion and rate of change between consecutive notes, any two consecutive notes that is.  That’s where math comes in to save the day.  Let’s build the rate of change, shall we.

First, note that the rate is increasing, at an increasing rate, so we cannot add.  I show that in the video below.  We have to multiply.  When we repeatedly multiply, we can use exponents.  Since we need a note and it’s octave to be doubles, our base number is 2.

Since there are twelve notes between a note and its octave, we need to break the multiple of two into twelve equal, multiplicative parts.  That’s a rational exponent, 1/12.

The number we need to multiply each note by is 21/12.  Each note is one-twelfth of the way to the octave.  It is pretty cool indeed.

For a more in-exploration, visit this page.

For a great read on this topic, consider the book Harmonograph.

Rational Exponents and Logarithmic Counting …

For a PDF copy of the following, please click here.

rational exponents

Rational Exponents

In the last section we looked at some expressions like, “What is the third root of twenty-seven, squared?” The math is kind of ugly looking.

27 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aIYaGaaG4namaaCaaaleqabaGaaGOmaaaaaeaacaaIZaaaaaaa@392A@

The procedures are clunky and it is very easy to lose sight of the objective. What this expression is asking is what number cubed is twenty-seven squared. You could always square the 27, to arrive at 729 and see if that is a perfect cube.

There is a much more elegant way to go about this type of calculation. Turns out if we rewrite this expression with a rational exponent, life gets easier.

27 2 3 = 27 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aIYaGaaG4namaaCaaaleqabaGaaGOmaaaaaeaacaaIZaaaaOGaeyyp a0JaaGOmaiaaiEdadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaaaa a@3E10@

These two statements are the same. They ask the same question, what number cubed is twenty-seven squared?

By now you should be familiar with perfect cubes and squares. Hopefully you’re also familiar with higher powers of 2 and 3, as well as a few others. For example, you should recognize that 625 is 5 4 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaCa aaleqabaGaaGinaaaakiaac6caaaa@385D@ If you don’t know that yet, a cheat sheet might be helpful.

Let’s look at our expression again. If you notice that 27 is a perfect cube, then you can rewrite it like this:

27 2/3 ( 3 3 ) 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaiE dadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaakiabgkziUoaabmaa baGaaG4mamaaCaaaleqabaGaaG4maaaaaOGaayjkaiaawMcaamaaCa aaleqabaGaaGOmaiaac+cacaaIZaaaaaaa@4157@

Maybe you see what’s going to happen next, but if not, we have a power raised to another here, we can multiply those exponents. Three times two-thirds is two. This becomes three squared.

( 3 3 ) 2/3 3 2 =9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaWaaWbaaSqabeaacaaIZaaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaGaai4laiaaiodaaaGccqGHsgIRcaaIZaWaaWbaaSqabe aacaaIYaaaaOGaeyypa0JaaGyoaaaa@40FA@

Not too bad! We factor, writing the base of twenty-seven as an exponent with a power that matches the denominator of the other exponent, multiply, reduce, done!

 

 

Let’s look at another.

Simplify:

625 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaik dacaaI1aWaaWbaaSqabeaacaaIZaGaai4laiaaisdaaaaaaa@3A8D@

We mentioned earlier that 625 was a power of 5, the fourth power of five. That’s the key to making these simple. Let’s rewrite 625 as a power of five.

( 5 4 ) 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI1aWaaWbaaSqabeaacaaI0aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIZaGaai4laiaaisdaaaaaaa@3B8F@

We can multiply those exponents, giving us five-cubed, or 125. Much cleaner than finding the fourth root of six hundred and twenty-five cubed.

What about something that doesn’t work out so, well, pretty? Something where the base cannot be rewritten as an exponent that matches the denominator?

32 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaik dadaahaaWcbeqaaiaaiodacaGGVaGaaGinaaaaaaa@39CB@

This is where proficiency and familiarity with powers of two comes to play. Thirty-two is a power of two, just not the fourth power, but the fifth.

( 2 5 ) 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIYaWaaWbaaSqabeaacaaI1aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIZaGaai4laiaaisdaaaaaaa@3B8D@

If we multiplied these exponents together we end up with something that isn’t so pretty, 2 15/4 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGymaiaaiwdacaGGVaGaaGinaaaakiaac6caaaa@3A87@ We could rewrite this by simplifying the exponent, but there’s a better way. Consider the following, and note that we broke the five twos into a group of four and another group of one.

( 2 5 ) 3/4 = ( 2 1 2 4 ) 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIYaWaaWbaaSqabeaacaaI1aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIZaGaai4laiaaisdaaaGccqGH9aqpdaqadaqaaiaaikdada ahaaWcbeqaaiaaigdaaaGccqGHflY1caaIYaWaaWbaaSqabeaacaaI 0aaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaGaai4laiaais daaaaaaa@462A@

Now we’d have to multiply the exponents inside the parenthesis by ¾ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=5laaaa@384E@ , and will arrive at:

2 3/4 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaG4maiaac+cacaaI0aaaaOGaeyyXICTaaGOmamaaCaaa leqabaGaaG4maaaaaaa@3D08@

Notice that 2 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaG4maiaac+cacaaI0aaaaaaa@390E@ is irrational, so not much we can do with it, but two cubed is eight. Let’s write the rational number first, and rewrite that irrational number as a radical expression:

8 2 3 4 ,or8 8 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioamaake aabaGaaGOmamaaCaaaleqabaGaaG4maaaaaeaacaaI0aaaaOGaaiil aiaaykW7caaMc8Uaam4BaiaadkhacaaMc8UaaGPaVlaaiIdadaGcba qaaiaaiIdaaSqaaiaaisdaaaaaaa@445B@ .

There’s an even easier way to think about these rational exponents. I'd like to introduce something called Logarithmic Counting.  For those who don't know what logarithms are, that might sound scary.

Do you remember learning how to multiply by 5s...how you'd skip count?  (5, 10, 15, 20, ...)  Logarithmic counting is the same way, except with exponents.  For example, by 2:  2, 4, 8, 16, 32, ... Well, what’s the fourth step of 2 when logarithmically counting? It’s 16, right? MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F6@

Let’s look at 16 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiA dadaahaaWcbeqaaiaaiodacaGGVaGaaGinaaaaaaa@39CD@ . See the denominator of four? That means we’re looking for a fourth root, a number times itself four times that equals 16. The three, in the numerator, it says, what number is three of the four steps on the way to sixteen?

2 4 8 16

Above is how we get to sixteen by multiplying a number by itself four times. Do you see the third step is eight?

Let’s see how our procedure looks:

Procedure 1:

16 3/4 = ( 2 4 ) 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiA dadaahaaWcbeqaaiaaiodacaGGVaGaaGinaaaakiabg2da9maabmaa baGaaGOmamaaCaaaleqabaGaaGinaaaaaOGaayjkaiaawMcaamaaCa aaleqabaGaaG4maiaac+cacaaI0aaaaaaa@4072@

( 2 4 ) 3/4 = 2 4 1 × 3 4 = 2 3 ,or8. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIYaWaaWbaaSqabeaacaaI0aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIZaGaai4laiaaisdaaaGccqGH9aqpcaaIYaWaaWbaaSqabe aadaWcaaqaaiaaisdaaeaacaaIXaaaaiabgEna0oaalaaabaGaaG4m aaqaaiaaisdaaaaaaOGaeyypa0JaaGOmamaaCaaaleqabaGaaG4maa aakiaacYcacaaMc8UaaGPaVlaad+gacaWGYbGaaGPaVlaaykW7caaI 4aGaaiOlaaaa@4FAB@

Procedure 2:

16 3/4 = 16 3 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiA dadaahaaWcbeqaaiaaiodacaGGVaGaaGinaaaakiabg2da9maakeaa baGaaGymaiaaiAdadaahaaWcbeqaaiaaiodaaaaabaGaaGinaaaaaa a@3E10@

16 3 4 = ( 2 4 ) 3 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aIXaGaaGOnamaaCaaaleqabaGaaG4maaaaaeaacaaI0aaaaOGaeyyp a0ZaaOqaaeaadaqadaqaaiaaikdadaahaaWcbeqaaiaaisdaaaaaki aawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaabaGaaGinaaaaaaa@3F2C@

( 2 4 ) 3 4 = 2 4 4 × 2 4 4 × 2 4 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaada qadaqaaiaaikdadaahaaWcbeqaaiaaisdaaaaakiaawIcacaGLPaaa daahaaWcbeqaaiaaiodaaaaabaGaaGinaaaakiabg2da9maakeaaba GaaGOmamaaCaaaleqabaGaaGinaaaaaeaacaaI0aaaaOGaey41aq7a aOqaaeaacaaIYaWaaWbaaSqabeaacaaI0aaaaaqaaiaaisdaaaGccq GHxdaTdaGcbaqaaiaaikdadaahaaWcbeqaaiaaisdaaaaabaGaaGin aaaaaaa@479A@

2 4 4 × 2 4 4 × 2 4 4 =2×2×2,or8. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aIYaWaaWbaaSqabeaacaaI0aaaaaqaaiaaisdaaaGccqGHxdaTdaGc baqaaiaaikdadaahaaWcbeqaaiaaisdaaaaabaGaaGinaaaakiabgE na0oaakeaabaGaaGOmamaaCaaaleqabaGaaGinaaaaaeaacaaI0aaa aOGaeyypa0JaaGOmaiabgEna0kaaikdacqGHxdaTcaaIYaGaaiilai aaykW7caaMc8Uaam4BaiaadkhacaaMc8UaaGPaVlaaykW7caaI4aGa aiOlaaaa@54D0@

The most elegant way is to realize the 16 is the fourth power of 2, and the fraction ¾ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=5laaaa@384E@ is asking us for the third entry. What is 3/4s of the way to 16 when multiplying (exponents)?

Let’s look at 625 2/3 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccaGGUaaa aa@3B47@ Let’s do this three ways, first with radical notation, then by evaluating the base and simplifying the exponents, and then by thinking about what is two thirds of the way to 625.

Now this is going to be a tricky problem because 625 is NOT a perfect cube. It is the fourth power of 5, though, which means that 125 (which is five-cubed) times five is 625.

Radical Notation:

625 2/3 = 625 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGH9aqp daGcbaqaaiaaiAdacaaIYaGaaGynamaaCaaaleqabaGaaGOmaaaaae aacaaIZaaaaaaa@3F8C@

625 2 3 = ( 5 4 ) 2 3 5 8 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aI2aGaaGOmaiaaiwdadaahaaWcbeqaaiaaikdaaaaabaGaaG4maaaa kiabg2da9maakeaabaWaaeWaaeaacaaI1aWaaWbaaSqabeaacaaI0a aaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaqaaiaaioda aaGccqGHsgIRdaGcbaqaaiaaiwdadaahaaWcbeqaaiaaiIdaaaaaba GaaG4maaaaaaa@445D@

5 8 3 = 5 3 × 5 3 × 5 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aI1aWaaWbaaSqabeaacaaI4aaaaaqaaiaaiodaaaGccqGH9aqpdaGc baqaaiaaiwdadaahaaWcbeqaaiaaiodaaaGccqGHxdaTcaaI1aWaaW baaSqabeaacaaIZaaaaOGaey41aqRaaGynamaaCaaaleqabaGaaGOm aaaaaeaacaaIZaaaaaaa@438B@

5 3 × 5 3 × 5 2 3 =5×5 25 3 ,or25 25 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aI1aWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaGynamaaCaaaleqa baGaaG4maaaakiabgEna0kaaiwdadaahaaWcbeqaaiaaikdaaaaaba GaaG4maaaakiabg2da9iaaiwdacqGHxdaTcaaI1aWaaOqaaeaacaaI YaGaaGynaaWcbaGaaG4maaaakiaacYcacaaMc8UaaGPaVlaaykW7ca WGVbGaamOCaiaaykW7caaMc8UaaGPaVlaaikdacaaI1aWaaOqaaeaa caaIYaGaaGynaaWcbaGaaG4maaaaaaa@56AD@

Pretty ugly!

Exponential Notation:

625 2/3 = ( 5 4 ) 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGH9aqp daqadaqaaiaaiwdadaahaaWcbeqaaiaaisdaaaaakiaawIcacaGLPa aadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaaaaa@4131@

( 5 4 ) 2/3 = ( 5 3 × 5 1 ) 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI1aWaaWbaaSqabeaacaaI0aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaGaai4laiaaiodaaaGccqGH9aqpdaqadaqaaiaaiwdada ahaaWcbeqaaiaaiodaaaGccqGHxdaTcaaI1aWaaWbaaSqabeaacaaI XaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaGaai4laiaaio daaaaaaa@45FA@

( 5 3 × 5 1 ) 2/3 = 5 2 × 5 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI1aWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaGynamaaCaaaleqa baGaaGymaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaiaac+ cacaaIZaaaaOGaeyypa0JaaGynamaaCaaaleqabaGaaGOmaaaakiab gEna0kaaiwdadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaaaaa@4745@

5 2 × 5 2/3 =25× 5 2/3 ,or25 25 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaCa aaleqabaGaaGOmaaaakiabgEna0kaaiwdadaahaaWcbeqaaiaaikda caGGVaGaaG4maaaakiabg2da9iaaikdacaaI1aGaey41aqRaaGynam aaCaaaleqabaGaaGOmaiaac+cacaaIZaaaaOGaaiilaiaaykW7caaM c8UaaGPaVlaad+gacaWGYbGaaGPaVlaaykW7caaMc8UaaGOmaiaaiw dadaGcbaqaaiaaikdacaaI1aaaleaacaaIZaaaaaaa@5447@

A little better, but still a few sticky points.

Now our third method.

625 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaaaaa@3A8B@ asks, “What is two thirds of the way to 625, for a cubed number?”

This 625 isn’t cubed, but a factor of it is.

625 2/3 = ( 125×5 ) 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGH9aqp daqadaqaaiaaigdacaaIYaGaaGynaiabgEna0kaaiwdaaiaawIcaca GLPaaadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaaaaa@4489@
This could also be written as:

125 2/3 × 5 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGHxdaT caaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaaaaa@3FBF@

I am certain that 5 to the two-thirds power is irrational because, well, five is a prime number. Let’s deal with the other portion.

The steps to 125 are: 5 25 125

The second step is 25.

125 2/3 × 5 2/3 =25× 5 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGHxdaT caaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGH9aqpca aIYaGaaGynaiabgEna0kaaiwdadaahaaWcbeqaaiaaikdacaGGVaGa aG4maaaaaaa@4779@

To summarize the denominator of the rational exponent is the index of a radical expression. The numerator is an exponent for the base. How you tackle the expressions is entirely up to you, but I would suggest proficiency in multiple methods as sometimes the math lends itself nicely to one method but not another.

 


Practice Problems:

MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F6@ Simplify the following: 1.   ( 16 x 16 ) 3/4 2. 128 5/6 3. 125 3 5 4. 32 3/5 5. ( 81 x 27 ) 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGtb GaaeyAaiaab2gacaqGWbGaaeiBaiaabMgacaqGMbGaaeyEaiaabcca caqG0bGaaeiAaiaabwgacaqGGaGaaeOzaiaab+gacaqGSbGaaeiBai aab+gacaqG3bGaaeyAaiaab6gacaqGNbGaaeOoaaqaaiaabgdacaqG UaGaaeiiaiaabccadaqadaqaaiaaigdacaaI2aGaamiEamaaCaaale qabaGaaGymaiaaiAdaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaa iodacaGGVaGaaGinaaaaaOqaaaqaaaqaaiaaikdacaGGUaGaaGPaVl aaykW7caaMc8UaaGymaiaaikdacaaI4aWaaWbaaSqabeaacaaI1aGa ai4laiaaiAdaaaaakeaaaeaaaeaaaeaacaaIZaGaaiOlaiaaykW7ca aMc8UaaGPaVpaakeaabaGaaGymaiaaikdacaaI1aWaaWbaaSqabeaa caaIZaaaaaqaaiaaiwdaaaaakeaaaeaaaeaaaeaacaaI0aGaaiOlai aaykW7caaMc8UaaGPaVlaaiodacaaIYaWaaWbaaSqabeaacaaIZaGa ai4laiaaiwdaaaaakeaaaeaaaeaaaeaacaaI1aGaaiOlaiaaykW7ca aMc8+aaeWaaeaacaaI4aGaaGymaiaadIhadaahaaWcbeqaaiaaikda caaI3aaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaGaai4lai aaiodaaaaaaaa@80AF@

Cube Roots and Higher Order Roots

other roots

Cube Roots
and
Other Radicals

Square roots ask what squared is the radicand. A geometric explanation is that given the area of a square, what’s the side length? A geometric explanation of a cube root is given the volume of a cube, what’s the side length. The way you find the volume of a cube is multiply the length by itself three times (cube it).

The way we write cube root is similar to square roots, with one very big difference, the index.

a squareroot a 3 cuberoot MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaGcaa qaaiaadggaaSqabaGccqGHsgIRcaaMc8UaaGPaVlaadohacaWGXbGa amyDaiaadggacaWGYbGaamyzaiaaykW7caaMc8UaamOCaiaad+gaca WGVbGaamiDaaqaamaakeaabaGaamyyaaWcbaGaaG4maaaakiabgkzi UkaaykW7caaMc8Uaam4yaiaadwhacaWGIbGaamyzaiaaykW7caaMc8 UaamOCaiaad+gacaWGVbGaamiDaaaaaa@5A15@

There actually is an index for a square root, but we don’t write the two. It is just assumed to be there.

Warning: When writing cube roots, or other roots, be careful to write the index in the proper place. If not, what you will write will look like multiplication and you can confuse yourself. When writing by hand, this is an easy thing to do.

3 8 8 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaka aabaGaaGioaaWcbeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVpaakeaabaGaaGioaaWcbaGaaG4maa aaaaa@4718@

To simplify a square root you factor the radicand and look for the largest perfect square. To simplify a cubed root you factor the radicand and find the largest perfect cube. A perfect cube is a number times itself three times. The first ten are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000.

Let’s see an example:

Simplify:

16 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aIXaGaaGOnaaWcbaGaaG4maaaaaaa@384A@

Factor the radicand, 16, find the largest perfect cube, which is 8.

8 3 × 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aI4aaaleaacaaIZaaaaOGaey41aq7aaOqaaeaacaaIYaaaleaacaaI Zaaaaaaa@3B46@

The cube root of eight is just two.

2 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaake aabaGaaGOmaaWcbaGaaG4maaaaaaa@3847@

The following is true,

16 3 =2 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aIXaGaaGOnaaWcbaGaaG4maaaakiabg2da9iaaikdadaGcbaqaaiaa ikdaaSqaaiaaiodaaaaaaa@3BAA@ ,

only if

( 2 2 3 ) 3 =16 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIYaWaaOqaaeaacaaIYaaaleaacaaIZaaaaaGccaGLOaGaayzkaaWa aWbaaSqabeaacaaIZaaaaOGaeyypa0JaaGymaiaaiAdaaaa@3D4F@

Arithmetic with other radicals, like cube roots, work the same as they do with square roots. We will multiply the rational numbers together, then the irrational numbers together, and then see if simplification can occur.

( 2 2 3 ) 3 = 2 3 × ( 2 3 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIYaWaaOqaaeaacaaIYaaaleaacaaIZaaaaaGccaGLOaGaayzkaaWa aWbaaSqabeaacaaIZaaaaOGaeyypa0JaaGOmamaaCaaaleqabaGaaG 4maaaakiabgEna0oaabmaabaWaaOqaaeaacaaIYaaaleaacaaIZaaa aaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaa@43AC@

Two cubed is just eight and the cube root of two cubed is the cube root of eight.

2 3 × ( 2 3 ) 3 =8×( 2 3 )( 2 3 )( 2 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaG4maaaakiabgEna0oaabmaabaWaaOqaaeaacaaIYaaa leaacaaIZaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaO Gaeyypa0JaaGioaiabgEna0oaabmaabaWaaOqaaeaacaaIYaaaleaa caaIZaaaaaGccaGLOaGaayzkaaWaaeWaaeaadaGcbaqaaiaaikdaaS qaaiaaiodaaaaakiaawIcacaGLPaaadaqadaqaamaakeaabaGaaGOm aaWcbaGaaG4maaaaaOGaayjkaiaawMcaaaaa@4B2C@

2 3 × ( 2 3 ) 3 =8× 222 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaG4maaaakiabgEna0oaabmaabaWaaOqaaeaacaaIYaaa leaacaaIZaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaO Gaeyypa0JaaGioaiabgEna0oaakeaabaGaaGOmaiabgwSixlaaikda cqGHflY1caaIYaaaleaacaaIZaaaaaaa@4957@

2 3 × ( 2 3 ) 3 =8× 8 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaG4maaaakiabgEna0oaabmaabaWaaOqaaeaacaaIYaaa leaacaaIZaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaO Gaeyypa0JaaGioaiabgEna0oaakeaabaGaaGioaaWcbaGaaG4maaaa aaa@4351@

The cube root of eight is just two.

8× 8 3 =8×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiabgE na0oaakeaabaGaaGioaaWcbaGaaG4maaaakiabg2da9iaaiIdacqGH xdaTcaaIYaaaaa@3F0E@

( 2 2 3 ) 3 =16 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIYaWaaOqaaeaacaaIYaaaleaacaaIZaaaaaGccaGLOaGaayzkaaWa aWbaaSqabeaacaaIZaaaaOGaeyypa0JaaGymaiaaiAdaaaa@3D4F@

Negatives and cube roots: The square root of a negative number is imagery. There isn’t a real number times itself that is negative because, well a negative squared is positive. Cubed numbers, though, can be negative.

3×3×3=27 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4maiabgEna0kabgkHiTiaaiodacqGHxdaTcqGHsislcaaIZaGaeyyp a0JaeyOeI0IaaGOmaiaaiEdaaaa@4293@

So the cube root of a negative number is, well, a negative number.

27 3 =3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaacq GHsislcaaIYaGaaG4naaWcbaGaaG4maaaakiabg2da9iabgkHiTiaa iodaaaa@3BF3@

Other indices (plural of index): The index tells you what power of a base to look for. For example, the 6th root is looking for a perfect 6th number, like 64. Sixty four is two to the sixth power.

64 6 =2because 2 6 =64. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aI2aGaaGinaaWcbaGaaGOnaaaakiabg2da9iaaikdacaaMb8UaaGza VlaaykW7caaMc8UaaGPaVlaadkgacaWGLbGaam4yaiaadggacaWG1b Gaam4CaiaadwgacaaMc8UaaGPaVlaaykW7caaIYaWaaWbaaSqabeaa caaI2aaaaOGaeyypa0JaaGOnaiaaisdacaGGUaaaaa@51D6@

A few points to make clear.

·         If the index is even and the radicand is negative, the number is irrational.

·         If the radicand does not contain a factor that is a perfect power of the index, the number is irrational

·         All operations, including rationalizing the denominator, work just as they do with square roots.

Rationalizing the Denominator:

Consider the following:

9 3 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI5aaabaWaaOqaaeaacaaIZaaaleaacaaIZaaaaaaaaaa@385F@

If we multiply by the cube root of three, we get this:

9 3 3 3 3 3 3 = 9 3 3 9 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI5aaabaWaaOqaaeaacaaIZaaaleaacaaIZaaaaaaakiabgwSixdbb Opaaaaaasvgza8qadaWcaaqaamaakeaabaGaaG4maaWcbaGaaG4maa aaaOqaamaakeaabaGaaG4maaWcbaGaaG4maaaaaaGcpaGaeyypa0Za aSaaaeaacaaI5aWaaOqaaeaacaaIZaaaleaacaaIZaaaaaGcbaWaaO qaaeaacaaI5aaaleaacaaIZaaaaaaaaaa@454D@

Since 9 is not a perfect cube, the denominator is still irrational. Instead, we need to multiply by the cube root of nine.

9 3 3 9 3 9 3 = 9 9 3 27 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI5aaabaWaaOqaaeaacaaIZaaaleaacaaIZaaaaaaakiabgwSixdbb Opaaaaaasvgza8qadaWcaaqaamaakeaabaGaaGyoaaWcbaGaaG4maa aaaOqaamaakeaabaGaaGyoaaWcbaGaaG4maaaaaaGcpaGaeyypa0Za aSaaaeaacaaI5aWaaOqaaeaacaaI5aaaleaacaaIZaaaaaGcbaWaaO qaaeaacaaIYaGaaG4naaWcbaGaaG4maaaaaaaaaa@4619@

Since twenty seven is a perfect cube, this can be simplified.

9 3 3 = 9 9 3 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI5aaabaWaaOqaaeaacaaIZaaaleaacaaIZaaaaaaakiabg2da9maa laaabaGaaGyoamaakeaabaGaaGyoaaWcbaGaaG4maaaaaOqaaiaaio daaaaaaa@3CA4@

And always make sure to reduce if possible.

9 3 3 =3 9 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI5aaabaWaaOqaaeaacaaIZaaaleaacaaIZaaaaaaakiabg2da9iaa iodadaGcbaqaaiaaiMdaaSqaaiaaiodaaaaaaa@3BC7@

This is a bit tricky, to be sure. The way the math is written does not offer us a clear insight into how to manage the situation. However, the topic we will see next, rational exponents, will make this much clearer.

 

Practice Problems:


Simplify or perform the indicated operations:

1. 64 4 2. 9 3 +4 9 3 3. 9 3 ×4 9 3 4. 64 5 5. 7 7 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIXa GaaiOlaiaaykW7caaMc8+aaOqaaeaacaaI2aGaaGinaaWcbaGaaGin aaaaaOqaaaqaaaqaaiaaikdacaGGUaGaaGPaVlaaykW7daGcbaqaai aaiMdaaSqaaiaaiodaaaGccqGHRaWkcaaI0aWaaOqaaeaacaaI5aaa leaacaaIZaaaaaGcbaaabaaabaaabaGaaG4maiaac6cacaaMc8UaaG PaVpaakeaabaGaaGyoaaWcbaGaaG4maaaakiabgEna0kaaisdadaGc baqaaiaaiMdaaSqaaiaaiodaaaaakeaaaeaaaeaacaaI0aGaaiOlai aaykW7caaMc8+aaOqaaeaacaaI2aGaaGinaaWcbaGaaGynaaaaaOqa aaqaaaqaaiaaiwdacaGGUaGaaGPaVlaaykW7caaMc8+aaSaaaeaada GcaaqaaiaaiEdaaSqabaaakeaadaGcbaqaaiaaiEdaaSqaaiaaioda aaaaaaaaaa@6089@

Multiplying and Dividing Square Roots, Rationalizing the Denominator

1.7.2 square root operations continued

Square Roots
Multiplication and Division

At some point square roots should no longer be considered an operation but rather the most efficient way to express a number. For example, the best way to write one hundred trillion is 1× 10 14 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgE na0kaaigdacaaIWaWaaWbaaSqabeaacaaIXaGaaGinaaaaaaa@3BE4@ . The best way to express the number times itself that is two is as 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIYaaaleqaaOGaaiOlaaaa@378A@

That provides insight when we consider multiplying a rational number and an irrational number together. It is not confusing for some irrational numbers, like π. Nobody confused 3π because we understand that symbol is the best way to write the number. There’s not a way to rewrite multiples of π other than by writing the multiple in front.

However, 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaka aabaGaaGOmaaWcbeaaaaa@378B@ is often written as 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI2aaaleqaaaaa@36D2@ . There are reasons explained by the order of operations which tell us why this is false, but understanding what the square root of two is perhaps offers the simplest insight.

2 1.414 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIYaaaleqaaOGaeyisISRaaGymaiaac6cacaaI0aGaaGymaiaaisda aaa@3C2D@

3 2 = 2 + 2 + 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaka aabaGaaGOmaaWcbeaakiabg2da9maakaaabaGaaGOmaaWcbeaakiab gUcaRmaakaaabaGaaGOmaaWcbeaakiabgUcaRmaakaaabaGaaGOmaa Wcbeaaaaa@3CF8@

3 2 1.414+1.414+1.414 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaka aabaGaaGOmaaWcbeaakiabgIKi7kaaigdacaGGUaGaaGinaiaaigda caaI0aGaey4kaSIaaGymaiaac6cacaaI0aGaaGymaiaaisdacqGHRa WkcaaIXaGaaiOlaiaaisdacaaIXaGaaGinaaaa@45F6@

4.242

The square root of six is approximately 2.449. Not the same thing at all.

 

The following, however, is true:

2 × 3 = 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIYaaaleqaaOGaey41aq7aaOaaaeaacaaIZaaaleqaaOGaeyypa0Za aOaaaeaacaaI2aaaleqaaaaa@3BB2@

and

2×3 = 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIYaGaey41aqRaaG4maaWcbeaakiabg2da9maakaaabaGaaGOnaaWc beaaaaa@3B8C@ .

The following generalization can be used. Sometimes it is best to write things one way versus another, and it is up to you to decide if rewriting an expression offers insight.

ab = a b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca WGHbGaamOyaaWcbeaakiabg2da9maakaaabaGaamyyaaWcbeaakiab gwSixpaakaaabaGaamOyaaWcbeaaaaa@3D46@

If two numbers are both square roots you can multiply their radicands together. But you cannot multiply the radicand of a square root with rational number like we saw above.

Division is a little more nuanced, but only when your denominator is a fraction.

This generalization is true for division:

a b = a b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaada WcaaqaaiaadggaaeaacaWGIbaaaaWcbeaakiabg2da9maalaaabaWa aOaaaeaacaWGHbaaleqaaaGcbaWaaOaaaeaacaWGIbaaleqaaaaaaa a@3B1C@

For example:

8 4 = 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada GcaaqaaiaaiIdaaSqabaaakeaadaGcaaqaaiaaisdaaSqabaaaaOGa eyypa0ZaaOaaaeaacaaIYaaaleqaaOGaaiOlaaaa@3A6A@

This can be calculated two ways.

8 4 = 8 4 = 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada GcaaqaaiaaiIdaaSqabaaakeaadaGcaaqaaiaaisdaaSqabaaaaOGa eyypa0ZaaOaaaeaadaWcaaqaaiaaiIdaaeaacaaI0aaaaaWcbeaaki abg2da9maakaaabaGaaGOmaaWcbeaakiaac6caaaa@3D25@

or

8 4 = 2 2 2 = 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada GcaaqaaiaaiIdaaSqabaaakeaadaGcaaqaaiaaisdaaSqabaaaaOGa eyypa0ZaaSaaaeaacaaIYaWaaOaaaeaacaaIYaaaleqaaaGcbaGaaG OmaaaacqGH9aqpdaGcaaqaaiaaikdaaSqabaGccaGGUaaaaa@3DD9@

But you cannot divide rational numbers into the radicand, or the radicand of a square root into a rational number. Remember, square roots, when simplified, are the most efficient way of writing irrational numbers. If we used k to represent the square root of two, these types of confusing things would not be happening.


Nobody would confuse what is happening with
6 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI2aaabaGaam4Aaaaaaaa@37B7@ . We simply cannot evaluate that because 6 and k do not have common factors. When k is written as the square root of two, sometimes people just see a 2 and reduce.

The only issue with division of square roots occurs if you end up with a square root in the denominator.

5 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaWaaOaaaeaacaaIYaaaleqaaaaaaaa@379D@

Denominators must be rational and the square root of two is irrational. However, there’s an easy fix. Remember that 2 × 2 = 4 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIYaaaleqaaOGaey41aq7aaOaaaeaacaaIYaaaleqaaOGaeyypa0Za aOaaaeaacaaI0aaaleqaaOGaaiilaaaa@3C69@ and 4 =2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI0aaaleqaaOGaeyypa0JaaGOmaiaac6caaaa@394E@ It is also true that:

2 2 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada GcaaqaaiaaikdaaSqabaaakeaadaGcaaqaaiaaikdaaSqabaaaaOGa eyypa0JaaGymaaaa@398A@ .

To Rationalize the Denominator, which means make the denominator a rational number, we just multiply as follows:

5 2 2 2 = 5 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaWaaOaaaeaacaaIYaaaleqaaaaakiabgwSixpaalaaabaWa aOaaaeaacaaIYaaaleqaaaGcbaWaaOaaaeaacaaIYaaaleqaaaaaki abg2da9maalaaabaGaaGynamaakaaabaGaaGOmaaWcbeaaaOqaaiaa ikdaaaaaaa@3F35@

Sometimes we end up with something like this:

5 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaaG4mamaakaaabaGaaGOmaaWcbeaaaaaaaa@385A@

Three is a rational number and is perfectly okay in the denominator. If you multiply by the fraction 3 2 3 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaOaaaeaacaaIYaaaleqaaaGcbaGaaG4mamaakaaabaGaaGOm aaWcbeaaaaGccaGGSaaaaa@39F3@ you can still get the simplified equivalent, but you’ll have extra reducing to do at the end. Instead, just multiply by the irrational portion.

5 3 2 2 2 = 5 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaaG4mamaakaaabaGaaGOmaaWcbeaaaaGccqGHflY1daWc aaqaamaakaaabaGaaGOmaaWcbeaaaOqaamaakaaabaGaaGOmaaWcbe aaaaGccqGH9aqpdaWcaaqaaiaaiwdadaGcaaqaaiaaikdaaSqabaaa keaacaaI2aaaaaaa@3FF6@ .

In summary, to divide or multiply with square roots, you can multiply or divide the radicands. However, if you’re multiplying or dividing rational numbers and square roots, you cannot combine the radicands and the rational numbers.

 

 

 

 

 

 

 

 

Practice Problems:

 

Perform the indicated operations:

1.( 5 7 )( 3 14 ) 2.( 15 )( 3 ) 3. 3 2 8 4. 5 3 5. 3 8 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIXa GaaiOlaiaaykW7caaMc8UaaGPaVpaabmaabaGaaGynamaakaaabaGa aG4naaWcbeaaaOGaayjkaiaawMcaamaabmaabaGaaG4mamaakaaaba GaaGymaiaaisdaaSqabaaakiaawIcacaGLPaaaaeaaaeaaaeaacaaI YaGaaiOlaiaaykW7caaMc8+aaeWaaeaadaGcaaqaaiaaigdacaaI1a aaleqaaaGccaGLOaGaayzkaaWaaeWaaeaadaGcaaqaaiaaiodaaSqa baaakiaawIcacaGLPaaaaeaaaeaaaeaacaaIZaGaaiOlaiaaykW7ca aMc8+aaSaaaeaacaaIZaWaaOaaaeaacaaIYaaaleqaaaGcbaWaaOaa aeaacaaI4aaaleqaaaaaaOqaaaqaaaqaaiaaisdacaGGUaGaaGPaVl aaykW7caaMc8+aaSaaaeaadaGcaaqaaiaaiwdaaSqabaaakeaadaGc aaqaaiaaiodaaSqabaaaaaGcbaaabaaabaaabaGaaGynaiaac6caca aMc8UaaGPaVpaalaaabaGaaG4maaqaamaakaaabaGaaGioaaWcbeaa aaGccqGHflY1daWcaaqaamaakaaabaGaaGOmaaWcbeaaaOqaaiaaiA daaaaaaaa@6678@