01 - Simplify Square Roots with Factor Trees in Algebra (Radical Expressions), Part 1

hello welcome back to algebra we're going to start a new batch of lessons that are mostly gonna focus on the concept of square roots cubed roots radicals and so on and so forth eventually culminating and how to simplify expressions that contain these square roots and other radicals and also solve equations that contain square roots and radicals and then much later on we'll talk about the concept of an imaginary number which also involves the concept of the square root so we have a lot of ground to cover but I can honestly say that this material is absolutely critical because the concept of square roots the concept of radicals and cube roots is just something that's never going to go away in your study of algebra but the good news is it's very very easy to understand this stuff and it's actually kind of fun it's a nice change of pace from just solving equations and factoring and you're mostly dealing with numbers in the beginning so it's it's quite easy so once we get the hang of it so what we're gonna do is this is lesson we'll be quite long I want to give you a lot of problems all in one place so that you can see the progression of what does it look like with numbers and square roots what does it look like as we go into variables with square roots and so on so the title of this lesson is simplifying square roots with factor trees so we're gonna learn where the square root is we're gonna learn how to simplify square roots we're gonna learn how to use a factor tree and at the very end of the lesson we're going to learn how to deal with square roots that involve variables and then in following lessons we'll do the cube roots and fourth roots and in throats and and properties and so on so let's just jump right into it we have the concept of a square root you probably know what the general idea is from more basic math but we're gonna dive into it in greater detail here so we have the concept of square root all right the general idea of a square root is very very easy to understand right the idea is if you have this symbol here we're gonna talk about this a lot more later this little check mark thing is called a radical and we put a number underneath it we can put actually anything we want underneath it we can have variables like x squared or giant expressions under the radical but in this case we just have the number 25 what we say let me give you a few quick examples and you'll see the general idea and then I want to generalize it with with something very important to understand here in just a second so the square root of 25 is e equal to the number five why is it equal to the number five so over here I'm gonna write that's because if we take the answer that we guessed which I know you don't really know yet but I'm just showing you here five if we square it what do we get we get 25 back so in general the square root of any number is you're trying to call it go figure out what number exists such that if you square it then you get what's underneath the radical sign so it's the opposite kind of of squaring it's really the way you should think about it and that will become useful later on right so remember I'm giving you a little bit of a preview in solving equations you if you want to add something to both so if you want to if you have something added to both sides you do the opposite which is to subtract it if you have something multiplied then to do the opposite you would do the opposite of that which is division right then we learned about okay we have multiplication we can do the opposite which is division we have division which is the opposite which is multiplication and so on so every mathematical operation has you can call it it's opposite for now all right use a loose terminology well when we have something squared the opposite of that is the square root so later on when we solve the equations we will be undoing the squaring by applying a square root so it's one of these things you're going to use it to solve the equations you're gonna use for everything because if you have an equation that involves a square oftentimes you'll use a square root to undo it why is it the opposite well because I tell you here that I know that five squared is 25 because five times five is 25 so then if I go the opposite way and say the square root of 25 the answer is five so you see how if I were to apply a square root to both sides of this thing I will undo it and get what I started with so let's take a few more examples what would be then the square root of nine what number would I have to multiply by itself to give me nine three is the only thing that works right what if I have the square root of 49 what number times itself gives you 49 right well that forty-nine should ring a bell seven times seven is forty-nine right so to kind of catch up here I could say because 3 squared is 9 that's the reason this is true because 7 squared is equal to 49 that's the reason this is so every time you see a square root you're thinking to yourself what times what gives me what is under here all right then we have the square root of 100 what times what would give me a 100 the answer is not the 100 answer would be 10 right let me write down this one and then we'll switch gears a little bit because 10 squared is equal to 110 times 10 is 100 now so far everything that we've put under the radical here has been what we call a perfect square they've all been carefully chosen numbers that have a square root that is a whole number but what if you had something different like the square root of 12 well you see you're looking for a number what times what gives me 12 well ok 3 times 3 gives me 9 that's not right 4 times 4 gives me 16 that's not right so there is no whole number that works it does not mean that 12 doesn't have a square root it just means that there is no whole number that is the square root of 12 so of course 12 has a square root right any positive number is going to have a square root but that square root will not be a whole number unless you have these special cases up here so 12 has an answer but for right now I'm gonna say because I'm gonna show you how to do that later this is not a whole number because there is no number that I can multiply by itself to give me 12 now I'm gonna learn how to actually simplify that later same thing with for instance the square root of 18 you think ok 8 or at least say 4 times 4 is 16 okay that's not right 5 times 5 is 25 I've already blown it's right so this is not a whole number it does not mean that this doesn't exist it just means it's a fraction it's a decimal of some something and somewhere in between 4 and 5 because 4 times 4 is 16 5 times 5 is 25 so somewhere in the MB that range is going to be a number that would be the square root of 18 now we're gonna get to this much much later but but just for completeness what happens see all of these numbers are actually positive numbers right what happens if I do the square root of let's say negative 4 what is the square root of negative 4 you said well that's easy that's gonna be 2 right so you say well okay 2 times 2 well 2 times 2 is not naked for two times two is actually positive four so it can't be two so you put over here and that's not it right and you say well what can give me negative four well we'll try this negative two times negative two well actually that gives me also positive four so the answer cannot be negative two remember whenever you try to find the square root you're looking for a number that you multiply by itself that means it has to be the same thing either two times to try that or negative two times negative to try that neither one of them works right the only way that you can really get a negative four is by or one way you get a negative four is two times negative two but that that wouldn't satisfy a square root because we take a square root we want to find this the number that you can multiply by itself so it has to be identically multiplied by itself so there is no we call it there is no real number here that can give you a negative four so we say not a real number and you're looking at me cross-eyed and you're like what's a real number trust me we're gonna get into it later but it turns out that all of the numbers that you use in everyday life are real numbers you know five ten even the negative numbers those are real negative two negative twenty-five three point five how about fractions one-half negative one-half negative sixteen those are all real numbers right but it turns out that there are other numbers that we're going to learn how to deal with later those are called imaginary numbers so when we say the square root of negative four is not a real number we mean there is no real number that you can multiply to give me that but it turns out that there is an imaginary number we'll talk about it later that lets us solve this kind of problem but for now don't get so bogged down in that just know that you can't take the square root of a negative number at least to get a real answer at this stage of the game so as another example would be what if I have a square root of negative 36 a lot of students would say ah 36 that's gonna be 6 right well 6 times 6 is 36 right but that's not negative 36 so that's not gonna work what about negative 6 so negative 6 times negative 6 okay that's a still positive 36 so there's no way to get negative 36 with a real number so again you say not a real number but as a preview we will be able to take square roots of negative later on using what we call imaginary numbers and that's a crazy weird name but I don't like chili like that name very much but turns out imaginary numbers are really really useful just not right now let's get there when we get there all right now what I need to do before I go farther is I want to expand on this just a little bit more let's take this first example the square root of 25 being equal to 5 let me go over here and talk about something called the principal the principal is your pal the principal square root now what do I mean by that when we say when we have that problem that we just wrote down I just wrote down the answer was 5 so if I have the square root of 25 and I'm trying to solve that what does it mean we all know in words what it means we've been talking about it we're saying well what number so that you multiply it by itself gives us 25 that's what it's where it really is okay so in math terms what you really are trying to say is some number X I don't know what it is but when I square it mean when I multiply by itself I want it to equal 25 in words that's what I've been saying the whole time hey when I take the square root of this find a number so that I can multiply by itself and it will give me 25 that's what I've been saying so we said that X can equal 5 why because 5 times 5 is equal to 25 that's what we've been saying but it turns out that this equation here this one right here it turns out that there's another value of x that works can you guys guess what that value is X can also be negative 5 negative 5 actually also satisfies this equation forget about the square root business just say if solve this equation there are two answers the answer is 5 because 5 squared is 25 but the answer can be negative 5 why because negative 5 times negative 5 is also equal to negative times negative is positive 25 so when we talk about square roots it's a little bit confusing in the beginning but I promise you it will get easier as you go along but it's a little confusing because a lot of books throw this business of a principal square root in your face right at the beginning I didn't do that I said hey here's square root symbols we're trying to figure out the answers that make them work and everything was a positive number notice right it turns out that when you take the square root of a number you're really asking yourself what is it that satisfies this equation there are always two answers to that and what we say is that the positive answer this one we call this one the principal square root we call it the principal square root so in other words if I were solving this equation here is perfectly valid to solve this equation but this one the one that's a positive answer generally is the one unless we're solving equations down the road generally that's the one you're after that's the why when I wrote down these square root square roots on the board you know we're listening the positive values but they're technically called principle square roots so if I'm going to solve this equation x squared is equal to 25 really there are two values X can be equal to the square root of 25 this one was called the principal square root right but X can also be equal to a minus sign in front of the square root of 25 this one's just the other root because notice that's exactly what we had here we said the answer was 5 or negative 5 so the answer is 5 which is what we get when we do the square root of 25 or it's negative 5 which means negative sign in front of this thing which ends up being 5 so it's a little bit confusing in the beginning because a lot of students when they're taking square roots will say well what do I write down do I write down the positive 5 or do I write down the negative 5 here is your rule of thumb this is what I want you to do in this class and this is what I want you to do in your book and your homework and on through calculus physics and chemistry and so on ready here it is it's very important that you understand this all right here we go when you have a problem that asks you to find a square root of 25 and it put it has that radical symbol there with the 25 and it has a radical symbol find me the square root of 9 and it has that symbol find me the R at the square root of 49 and it says find me the square root of 100 and it says find me the square root of 12 and those radical symbols are written down as part of your problem statement then they're asking you for is tell me the positive value that's gonna multiply there it's basically they're asking that is the principle square root all right so pretty much in any problem where a radical is in your problem statement they're just asking you for the principal square root they don't want you to think about could it be negative could it be positive they just want you to write down the principal square root so if the problem says find the square root of 39,000 they just want to know what positive number multiplies by itself to give you 39,000 that's what you write down but later on we're gonna start solving equations so if I give you an equation that says x squared is equal to 25 solve this equation right then you solve this equation then I'm going to have to in order to get X by itself I told you remember square roots and squares are opposites of one another so just like whenever I'm solving an equation I might have to do the opposite I might have to do addition to get rid of subtraction or I might have to do multiplication to undo division if I cover all this stuff up and I say solve this equation notice there's no radical symbols in this equation I have been given an equation to solve but I don't have any radicals in this problem statement but in that case to get X by itself I'll have to undo the squaring I'll have to apply a square root to the left which will undo the squaring and then I'll have to apply a square root to the right to be able to take the square root whenever I do the action of applying a square root by myself as part of solving an equation then I will have to write down plus or minus there's two answers a plus square root of 25 the principal square root or a minus square root of 25 so that's a little bit front-loaded here I want to get that out of the way up front we're not gonna solve equations anytime soon we have several lessons down the road before we get to solving equations that involve radicals but the big overarching picture is if I give you a square root it has a radical written down all I want is to know the principal square root the positive values that multiply to give me there what's under that radical but if I give you an equation to solve there's no radicals written in that equation but you might have to apply a square root yourself to both sides then you have to give me both answers because I know there's two answers to this equation because there's a square remember when we talked about equations the highest power of the variable is the number of solution you have so you have to have two answers plus and minus five both work to satisfy that equation that was a lot of talking now we're gonna get back to what I think is a little more fun we're gonna talk about factor trees we're gonna talk about solving these things with factor trees factor trees now in these cases over here what I did as I said what is the square root of 25 we just guessed we knew what the answer was because we know our multiplication tables right but what I want to do is go back and do some of these same ones again but not using our multiplication tables I want to show you what a factor tree is so we're gonna do really easy problems first but the factor trees will be used to solve the much more complicated problems for instance the square root of 12 you'll be using a factor tree to do that the square root of 18 you'll be using a factor tree to do that so let's go back down and solve the same problems here that we did in the beginning but we'll use this technique which is I think one of the most powerful things you can learn square root of 25 we know it's 5 of course we do and we're just going to give the principal square root because the radical is already written now it's asking us for the principal square root here's what you do you draw a little tree here and you write down anything you can think of to multiply to give you 25 it doesn't have to be 5 times 5 you can think like when the problems get more complicated it'll there'll be more choices right now the only thing I can think of to multiply to give me 25 is 5 times 5 so I'm gonna write 5 I'm gonna put a dot here that means multiplication times 5 you see I'm building a tree of factors factor is something that multiplies to give me the number now because and this is important because I am taking a square root square root of you think of the number 2 right when you think of squaring you think of the number 2 these are square roots so because I'm looking for square for the square root of 25 when I write my factor tree I'm looking for any duplicate factors but in pairs any time I find a pair of them and it has to be a pair but if I find a pair of them I circle them and this pair only counts one time in the final answer so the square root of 25 means I found a pair I bring it out only one time in the answer is 5 now you already know the answer is 5 because 5 times 5 is 25 I'm showing you for a simple problem but I give you the square root of 1039 you're not gonna know what multiply you're not gonna know that you're gonna have to write a more complicated factor tree this is the general way to do it write your factors look for pairs when you find a pair you only count at one time you pull it out boom that's the answer so let's crank through now that you know the concept of the factor tree let's go take a look at the square root of nine you all know that nine can be written as what three times three that's the only thing I can really think of anyway of course I could do one times nine but then that would have to break up the 9 into three times three and it would be the same thing so I have three times three here and I'm gonna circle this as a pair because it only counts as one unit I take that guy out and so the answer is three because it only counts one time I should say here in the beginning when we're building our factor trees you can think of anything you want to multiply under the radical you know for instance if I'm doing the square root of you know let's say 40 there's lots of ways to multiply to give me 40 right 10 times 4 is 40 8 times 5 is 40 so I can pick anything I want in my tree but I have to keep getting smaller and smaller building my tree out until all I have at the bottom of the tree is prime numbers remember a prime number is a number that can only be divided by itself and then number 1 so 3 is prime because I can't factor it any further well if I do I'm just gonna get 1 times 3 which are still still prime right 5 is prime because of course I can write 1 times 5 that's the only thing I can do to give me 5 but those are prime numbers so I basically stop when all I have at the bottom is is these these prime numbers let's do some more we're gonna do a lot more actually because I want to see I want you to see a lot of them all in one place what if we have 49 and we're gonna take the square root of 49 you think yourself what times what gives me 49 the first thing that comes into your mind is 7 times 7 is 49 this is a square root which means I'm looking for pairs so the 7 comes out one time and so the answer is 7 that's how you know the answer is 7 of course you know 7 squares 49 what if you have the number 100 taking the square root of that this is the same problems we did over there you think what times what gives me a hundred now here's a here's interesting thing you know that 10 times 100 right so if I were to do it as 10 let's do actually both ways just so you can see let's make it 10 times 10 all right now I said you go all the way down till you have primes numbers now 10 is not a prime number but it's it's perfect basically 10 times 10 as soon as I see a pair I can just stop and circle that pair and pull it out as 10 so the answer is 10 there but the number 100 let me do it right in the middle I don't like usually doing that but I don't want to go off to the other board here the number 100 right you can think of lots of things to give you 100 right what would be it another idea you could say 2 times 50 that gives you 100 right but 50 is now 2 cannot be broken down any more because 1 times 2 is 2 but 50 can be done as 2 times 25 right that gives you 50 now I'm starting to build a tree see 5 obviously is 5 times 5 so I have a nice perfect square here and I have a nice perfect square here of duplicates now here's the deal I'm looking for pairs so the 5 times 5 obviously is a pair so I Circle it the 2 times 2 that's in the bottom of the tree that counts as a pair 2 so when you're looking for these pairs all you're looking for is at the bottom of your tree you're skimming the bottom of the tree looking everywhere for any time there's a duplicate when you find it you circle a pair but you only count at one time so now we have a 2 which we can pull out to the answer but we also have this 5 that we can pull out to the answer which means the answer really consists of 2 times 5y times because this is a factor tree what you're basically have said is that 2 times 2 times 5 times 5 is what equals 100 so all of these things in the bottom of the tree are multiplied together think about it 2 times 2 is 4 4 times 5 is 20 20 times 5 is 100 so everything in the bottom of the tree is multiplied to give me 100 so when I take the square root the two counts once the 5 counts once but they're multiplied together and that means the answer is a hundred exactly the same thing as when we just happen to notice that 10 times 10 is 100 so we circle and stop right there so I guess my punchline here that I want you to understand is that when you're building your factor tree it doesn't matter what you choose like should I do 8 times 4 to be 32 or should I do something times something else to equal 32 it doesn't matter build your tree out no matter what you pick as long as you know your multiplication tables you will always get the correct value no matter which way you go in your tree when you're finding the square roots so here's a good example of that a little more complicated example slightly anyway let's say I have 225 and I'm taking the square root of that now I'm telling you right now there is a perfect square and some of you might look at this and know what the perfect square is but a lot of you certainly you know when I was taking this first time I did not know what what times itself gives me 225 now I know but back then I didn't that's okay if you don't you just think what times what will give me 225 anything times anything will give me 225 so I know it's divisible by five because there's five here so I get a calculator out and I figure out that five times 45 is equal to 225 how do I know that as I grab a calculator I take 225 I divided by 5 and I get 45 so I know that these have to multiply to give me this now five is prime I can't go any farther but I do know that 45 is five times nine or nine times five however you want to think about it again I have another five so I stopped there but the nine can be written as three times three so I've written a factor tree there now I look for pairs the three is a pair and the fives here are a pair so when I'm building my final answer I go over here and I said the answer will be this but counted only one time times this which only counts one time because I'm looking for pairs 5 times 3 is 15 so even though you might look over here and say I don't know what the square root of 225 is it actually does have a perfect square if you take 15 times itself 15 times 15 in your calculator and hit the enter key you will get 225 it's a perfect square but you may not have known it was a perfect square by just looking at it so it doesn't matter bills your factor trees look for pairs and you will always get the right answer all right because 15 squared is 225 so let's look at one that's a little more complicated than that let's go over here and say instead of 225 let's say 256 what's the square root of that all right what's that I have no idea if that's a perfect square or not but I do know that this is an even number because it ends in a six so I'm gonna get my calculator out and I'm gonna take 256 divided by two and I'm gonna figure out that 2 times 128 is actually what 256 is equal I'm done here but I know this is even so I'm gonna get my calculator and I'm gonna figure out that 2 times 64 is equal to 128 again I'm done here I don't need to go any farther but this is even as well so I can do anything I want here to be 64 I know it's divisible by 2 of course I could do 2 times something I could figure that out but 64 rings a bell in my head it rings a bell in my head right because 64 seems like for my multiplication tables perfect 8 times 8 is 64 so I can do that if I want 8 2 times 8 is 64 all right now truthfully you actually can stop here right so let's go ahead and do that real quick and just prove to ourselves that it doesn't matter how we go the answer here if I so I'm not gonna circle them yet but if I circle the twos they will count only one time if I circle the eights they will count only one time and so the answer should be 16 grab a calculator 16 times 16 does equal to 56 but let's say you just were sleeping and you didn't really realize you had a pair here and what you would do then is you say well 8 can be written as 2 times 4 and then this 4 can be written as 2 times 2 but then you have the same thing this 8 can be written as 2 times 4 and this 4 can be written as 2 times 2 now you have a lot of twos and what you're doing is you're looking only in the bottom of the tree this is the bottom this is the bottom these are the bottom this one here is also the bottom and this is the bottom and you're looking for pairs you're saying here's a pair of twos and then you're going down here and you're saying cuz this is a pair of twos okay and you're going over here and you're saying okay this is a pair of twos because it's in the bottom of the tree and you're saying this is a pair of twos here so I don't actually have any other orphans in the bottom here right they're all basically accounted for so this one counts once this one counts once this one and this one also counts once like this 2 times 2 is 4 4 times 2 is 8 8 times 2 is 16 you see you get exactly the same thing so the answer is 16 so if you happen to get down so that you have a perfect pair go ahead and circle it but if you don't see that just keep on going with that tree you're always gonna get the right answer and that's why I like this method so much because it's kind of fun to build these trees and you always get the right answer and you know you'd kind of just do multiplication tables very simple now let's take a look at a weird a slightly weird problem I call it weird it's not that weird let's go over here and create some room it's not a long problem what would be the square root of 38 they say okay that's an easy easy answer it's a it's an even number it's divisible by two so I'll grab my calculator and I figure out that two times 19 is equal to 38 and I can't really keep going with the tree here because all I have is one times two and honestly I can't even continue on with the tree here because 19 is is only one times 19 that's the only thing that would work so actually the tree is complete I can't go any farther but I don't have any pairs so a lot of students will sit there and say well what do I do there's nothing there's no pair well if there's no pair then there's nothing to pull out and so the answer is just you just have to write it as a square root of 38 that's the final answer you can't simplify it any more it's like when you get a fraction down and you get one half well you can't go any simpler than one half there's no way to simplify it any more so you just say the answer is one half right same thing here when you build a factor tree and you don't have anything to do you just say well this is an exact number square root of 38 is some decimal value it's not a whole number but I can't do it anymore exactly than that so I say that that's that's it okay well squeeze in another one right underneath here let's say we have the square root of 40 how do we do that well we know that 40 can be written lots of ways but the thing that popped in my mind was 4 times 10 of course you could do 2 times 20 if you wanted to you can do other things if you want but for right now I'm doing four times ten and then I see that the 4 can be written as 2 times 2 sorry I'm kind of running into the previous solution then 4 is 2 times 2 but the 10 can be written as 2 times 5 now I have all prime numbers in the bottom right here and so I'm basically done I look for pairs I see a pair right here this is a pair but this 2 is not paired with anything else so it can't be circled and the 5 is not paired with anything else so it cannot be circled so what do you do you take the only pair you have and you pull it out but again only once it only counts once so you have a 2 and then underneath this radical exists all of the stuff that didn't get pulled out it's just still under the radical and you write it as 2 times 5 that's what remains under the radical which is 2 times the square root of 10 this is the final answer 2 times the square root of 10 and I know that you look at this and you're thinking well that's a weird answer 2 times the square root of 10 but that's just the way it goes sometimes when you have a fraction and you reduce it down to 3/8 and you're like can I go any farther and you're like no I can't 3/8 is really it so this radical square root of 40 comes out to 2 times the square root of 10 but I cannot simplify the square root of 10 anymore because there is no square root of 10 that is a pole number of course square root of 10 is a decimal I can get in my calculator to get a decimal but as far as an exact value square root of 10 written as radical 10 is an exact value so I'll leave it just like this now let's prove to ourselves that this is actually the answer so let's go down here right what we're saying when we do something like square root of 25 we're saying the answers 5 because 5 times 5 is 25 so let's take this answer here 2 times the square root of 10 we're saying that if we square this answer we should get 40 back so what happens here you have two things inside just like any term in here with an exponent you apply the exponent to both things so you have 2 squared times this square root of 10 squared see if the exponent just applies to this a plus 2 this exactly the same rules of exponents from before the only difference is you have 2 squared that's easy that's a 4 but then you have square root 10 squared and here's how here's when you start to have to get a little bit of comfortable with what I was kind of telling you before the opposite of a square root is a square and the opposite of a square is a square root so you can think of it as matter and antimatter when they come together they annihilate a better example of math would be addition and subtraction right if I add 3 and I want to undo it I got to subtract 3 and when I do that they basically cancel each other out so if I have a square root of 10 but then I square it that the square root and the 10 actually cancel and just just totally disappear and all you have left is just a 10 that's underneath it because you're undoing the square root what's 4 times 10 that's 40 that's exactly what's underneath here so the answer really is 2 times the square root of 10 because when I square the answer I recover exactly what is underneath there to begin with all right we have more not too much more but we have some more and I want to again do a lot of these so you can really get it get comfortable with it let's say I had 60 and I wanted to take the square root of 60 what would I do I'd build a factor tree as usual right what times what gives me 60 there's lots of different things I can do 6 times 10 for instance but I can I'm gonna do 3 times 20 that's gonna give me 60 the 3 is prime so I'm done with that but the 20 can be written many ways again I'm gonna do it as 2 times 10 the 2 is prime so I'm done with that now the 10 can be written as 2 times 5 and now I'm done because I have prime numbers in the bottom everywhere and I start looking for pairs the only pair that I have is the 2 so I circle that the 3 and the 5 have no pairs so what you do is you take the 2 out only once though underneath the radical lives what is still in the bottom of the tree multiplied 3 times 5 so you get 2 times the square root of 15 and that's the final answer how do you know that the answer really is 2 times the square root of 15 so let's check it if I take 2 times the square root of 15 and I square that thing what happens the exponent goes and applies to the 2 then the exponent goes and applies to the square root of 15 squared so I get 2 4 and then again the square root and the square poof they annihilate each other all you have left is the 15 in the middle 4 times 15 is 60 and that's why it is exactly equal to that so if you were like a genius with math you'd probably be able to figure this stuff out just by looking at it because you would know the square root of 15 would magically turn into what is needed to multiply by 4 and so on go backwards but most of us have to use a factor tree to figure out what the answers gonna be and that's why I'm teaching you this way all right couple more what if we have 75 take the square root of that okay I have lots of choices I'm going to do 25 times 3 that's prime so I'm done with that but the 25 can be written as 5 times 5 so then I look for pairs the 5 is a pair so it comes out as a single unit but then underneath is what is left over it's orphaned because it's a 3 here so the answer is 5 times the square root of 3 and that's the final answer and by the same thing if you square this the square would square the 5 making 25 the square would cancel the square root giving you 3 3 times the 25 gives you the 75 that's why that's the correct answer all right what if I have 90 I'm going to take the square root of 90 lots of choices here I could do 30 times 3 whatever I'm gonna do in this case 10 times 9 because that's what popped in my head the first time now 10 can be written as 2 times 5 and these are prime and then 9 came here in this 3 times 3 and the only pair I have is this one these guys have worked they're an orphan so the 3 comes out one time under the radical lives what is multiplied together 2 times 5 so I get 3 times the square root of 10 that is what the square root of 9 is equal to you want to check it squaring this in your mind the square will square the 3 giving you 9 the square will kill the square root giving you 10 10 times 9 is 90 that's exactly what it should be all right now here's where we get a little interesting I kind of mentioned this before but let's try to take the square root of a negative number and see what happens what if I take the square root of negative 36 how would I do that well we already talked about before that you can't take the square root of a negative number and get a real answer we talked about that before but let's just try right how can we get negative 36 it can't be 6 times 6 it can't be negative 6 times negative 6 because those will both give me positive answers so the only way it really works is if I do negative 6 times a positive 6 that will multiply to give me the negative 36 but these don't make a pair because one's negative and ones positive so I can't circle it I can't pull it out so the answer is the square root of 36 is just the squirt I'm sorry the square root of negative 36 is just what it is the square root of negative 36 you have no real answer which is what we said before I'm showing you what the factor tree that it makes sense is really all I'm trying to do here same thing with negative 49 if I take the square root of that I think well what times what will give me that I know 7 times 7 works but 7 times 7 will give me positive 49 so really the only way it works is if it's 7 times negative 7 that'll give me the negative 49 but these aren't a pair so really there's nothing I can pull out and so what you end up having is the square root of negative 49 is just the square root of negative 49 whatever that is but there's no real number that satisfies it so you say no real answer and later on we'll introduce a complex an imaginary number to talk to you about what that is okay all right so just a couple of additional ones I would like to do let me see where do I have the room here yeah we can fit it in here no problem it's not it's not a difficult thing all right so let's go here so so far every problem we've done has had numbers and that's fine I like numbers I like introducing things with numbers because everybody can think numbers and they understand numbers but now we're not going to get into a lot of detail but let's do a couple just a couple of problems that have a variable right so let's say you have the square root but underneath here you have a the variable a squared and you don't know what a is it could be negative a could be positive how would you solve that well we do the same thing we always do we try to build a factor tree a squared is a times a and we look for pairs of numbers so we circle this because this is a pair and we pull it out once and so the answer is a but there's a but here that I have to introduce here that we will get to a lot more detail later right but it is true they see what's going on here the square root kind of cancels sort of with the square giving me just kind of what's under there and we've kind of seen that kind of before with the square root and the square kind of cancel there right but we have to be a little bit careful so we have to say assume that a is greater than zero because here's the thing and this is kind of getting into the minutiae to be honest with you all I really want you to know is that the square and the square root cancel and then you recover the variable bath but to be more mathematically rigorous about it you have to be a little careful the problem is a is now a variable a can be any number it can be positive values of numbers it can be zero and a can also be negative numbers right so the problem arises is what happens if a actually ends up being negative like what if a right now it's it's just a variable but what if it is I'm beaming negative one or something will this actually work if a is negative let's just check it out real quick and see what happens if a is negative then let's say negative one let's say a is equal to negative one let's let just to check this let a is equal to negative one then what we're saying is the square root of a squared is negative one times negative one that's what a squared is all right but that means that underneath this radical negative one times negative one is positive one and the square root of one is positive one so you see what happens I let what we're saying by this by this little little factor tree thing is that the square root of a squared is equal to the variable itself but the problem is when we actually let a become negative we take the square root of it we got a positive value back so a is not equal to itself they're they're not equal so this whole thing here only works if a is bigger than zero let's take that case if we let a equals positive one then under the radical we'll have 1 times 1 because a is now equal to 1 so that means we'll have the square root of 1 which is equal to 1 so if we let a be a positive value we square it take the square root then we get the same thing back but if we let a become a negative number then we square it take the square root we actually get the opposite sign so when you're taking the square root of things that are squared and they're variables and they can be positive or negative then you can say that it's equal to cancel like this but you have to assume that the variable is greater than zero some books will emphasize that more and some books won't some problems will give you the answers where where a has to be absolute value of a right we'll talk I'm going to talk to you a little bit more of that about that that notation as we do some future lessons I'm actually going to introduce it a little bit more rigorously but for now just know that when you have variables under here if you take the square root of something squared you do have to worry a little bit because for all for negative and positive values don't always actually work out so sometimes you have absolute values running around okay so with that aside let's go over here and just do and wrap up the rest of this lesson there's only a couple of additional ones but I wanted to show a few different things what if I have X to the fourth power I'm gonna take the square root of that well I know that X to the fourth power can be written as x squared times x squared and here I have a pair so I can say this is equal to x squared all right and then what if I have X to the fifth power and I'm gonna take the square root of that well I do a factor tree right so let's do x times x times X times X times X that's what X to the fifth is x times X times X times X times X but look I have a pair here and I have a pair here but this one is unpaired so what I have is this X come out comes out one time multiplied by this X that comes out one time and underneath the radical is just the single X that's left over so what you actually have is x squared times the square root of x and that would be the final answer for that one all right and then the final problem I think I have a room right here I'm gonna write right over here this is our last problem hmm kind of give me some some room here X to the sixth take the square root of that same thing I can do x times X times X and I can do it six times but I also know that X cubed times X cubed will give me X to the six because I can add the exponents and then I have a pair immediately so I can stop so the answer here will be X cubed so you see if you have variables underneath the radical sign you build the factor tree the same way you look for pairs the same way nothing's different it's just that in some cases you do have to say with an absolute value that that the variable has to be greater than zero or you have to specify it with an absolute value sometimes I'm gonna get I'm gonna get into that a little bit more as we solve some more problems but for now I mostly wanted to introduce the concept of a square root a square root is saying that we know that something times itself will give me the number right when you have a radical in your problem statement they're only asking you for the principal square root so you just give them the positive answer but later on when we start solving equations x squared is equal to something where we have to apply the square root to both sides by our own hand to cancel the squaring of this variable here for instance then in that case we have to put down okay we have plus or minus we have two answers because really five squared is 25 and negative five times negative five also is positive 25 so make sure to understand this I know it was a longer lesson but I wanted to give you lots of problems all in one place so you could really see how it all ties together we're not done with square roots follow me on to the next lesson we'll get some more practice right now

Channel: Math and Science

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Keywords: square root, square roots, simplify square roots, simplify square roots with variables, simplify square root expressions, simplify square roots fractions, simplify square roots with variables and exponents, simplifying algebraic square roots, algebraic square roots, radical expressions, simplify radical expressions, simplify radical expressions with variables, simplify radical exponents, algebra, algebra 1, algebra 2, factor tree, factors, simplify expressions, simplify radicals

Id: SHaf0G_o-A4

Length: 43min 39sec (2619 seconds)

Published: Tue Apr 30 2019