all right two and one it's boogie so introduction to radicals and roots so this is a topic that sometimes will be covered either in elementary school depends if your teacher really pushes the envelope or in grade nine maybe possibly higher but you can't really get away from roots or sometimes they're called radicals just depends on who's teaching i primarily like to use the word roots but radicals i know is also commonly used so before i jump into trying to explain these this is just an introduction topic and i'll have a few videos to the list so i'll make a little playlist on these radicals and roots so i'll put the entire list link up above where i do simplifications and others once they're up so on the left hand side what you see is you basically see kind of exponents right so you have 2 squared so you know that's 2 times 2 and equals to 4 and then you have 2 cubed and 2 to the power of 4 and then so on until 2 to the power of n where n is just the number of times that we multiply 2 by itself now these exponents and exponent rules you should kind of know them okay before you jump into radicals i have done an actual full series on exponents so if you like i'll put a link up above there to the entire thing now so what that you have these kind of exponents right so you may remember that you have a base so this base is raised to some exponent and then it gives you an answer and you can go ahead and do that you know for higher and higher exponents and of course you know we could go as high as we like so for instance if i have say 2 and then so here's our exponent button x to the power of n all right and then if i did it even to the power of 10 you know you can go ahead and find your answers now sometimes you may not be allowed to use a calculator but you should know what that means so now let me now say the following what if we didn't know the base all right so here the base is 2 and then we're raising it to the power of 2 power of 3 power of 4. what if instead i wrote something like this now of course because i'm writing it side by side you know exactly what x is here so i have x into the 3 equal to 8 x to the 4 is equal to 16 and then so on x to the n now we don't know what n actually is but it will be some answer let's say y now when you look at this so you know x here is equal to 2 because you have the left hand side right beside you now when we do not know what the base is then we can find out what the actual base is by using radicals or by using roots so the equivalent way of writing the 2 to the power of 2 let's say equals to 4 you can write it in a different form so using a slightly different symbol so i'm going to show you okay what that symbol is so let me do that here and for that symbol so what you have is that your base x is actually equal to so you can take and now we use this this is called a radical or a root so this symbol so this is the radical or sometimes it's called the root and it is the root of your answer so notice your four basically goes under that radical and your exponent here so this exponent 2 will be placed right here all right so just above that little kick of a tail that you have there so this just is the second root of four or sometimes we say four radical two okay so i typically use roots so i'll say you know the second root of four now because this has become so common we typically don't write the two and we just write this you know and that's why you see that on calculators and people know that that is called the square root of because they just dropped this little 2 but the 2 is actually there and now if we would continue this for all of them so let's say for the second one so be x and now this would be the third root of eight all right so that's how we would write that now that three cannot be dropped anymore because we want to be able to say what root it is okay now the following one is going to be x is equal to 4 and then this is okay root of 16 so the fourth root of 16 that's how you would read this and then this continues on where basically you can find out any root that you like so this is the nth root in this case of y so that's what you have so that is the the radical and it basically just finds out the base for you so you know if you took out so let's say if i had a calculator here so you'll notice that on the calculator we have these roots or these radicals symbols there now depends on your calculator you know you may have so just a simple square root so for example square root of 4 you know if you hit this it's going to give you the base back which is 2. now remember what that means is 2 times 2 is equal to 4 which is going to give you what you have under the radical now this okay so when we have this four when it is placed under the radical we call this erratic hint so i know that this is kind of weird terminology but that's what it is so in the bottom one the eight is the radicand for the third root of eight and then sixteen is the radicand okay for the fourth root all right and you can continue this you can find all the answers now because i mean we already know what the exponents are in this case so it's 3 4 and so on and we know what the base is but you know you should also be able to use it on your calculator as well and if you're not just doing a square root then you have to look for a symbol so notice that on here so what you have is i have this symbol which looks like this on your calculator and then you have to learn how to input your actual numbers in and that will depend on your on your calculator so unfortunately you know it may not be exactly the same as here you have to learn your calculator how to do that so for instance if i wanted to take the let's say 4th root of 16 so i cannot press just square root of 16. now you may already know that the square root of 16 is 4. all right so here i have to use this and then i have to put what my n is so this is the fourth root so notice of 16 and it equals to 2. all right so that's how you utilize these buttons on your calculators and hopefully now you know you know the root or the radical okay and how to do it so most of the time when you're just starting these things teachers will force you not to use a calculator i hope and you kind of learn to simplify these things on your own using prime factorization and i will have a video in this series on that as well okay so you can go back to the playlist that i linked in the beginning of the video and you know you can go and find the simplifications how to do that without any calculators so i'm not going to do that here because i just want to introduce the topic okay and introduce this root or this radical so now that you can kind of see what these radicals or roots actually do they basically are just finding whatever your base is okay so whatever your base is so notice on the left hand side okay we can find i mean for all of these it's always equal to two because that's exactly what i have set them out to be but they're of course not just with two they can be with anything so for instance if you take okay so if i take the square root of nine now this of course we would just simply write as that again this two is dropped from here it doesn't mean that it's dropped and it all of a sudden is equal to 1 it's not it's still equal to 2 and the square root of 9 is just simply to 3 and that's because 3 squared is equal to 9 which is your radicand all right under the radical so you can always double check in that way so you're doing kind of a reverse now and it's not just for square roots it's obviously for others so for instance if we had the cubed root of 27 you'll notice that this also is equal to 3 all right because again so 3 to the power of 3 is equal to 27 and that's exactly what our radicand is okay underneath that radical sign so that's what we have now they do become more and more complicated all right so i'm gonna just show you here so some other examples so here's four examples right here and you can see okay that these are becoming harder and they don't have to have the radicand so the radicand which is underneath right here they don't have to always be just integer numbers all right or just whole numbers so you'll notice that you have these right there and those are decimals the the third and fourth example the one thing that you do have to be careful especially as you're starting is that the radicand cannot be um negative all the time all right so sometimes it can be and sometimes it cannot be okay but when you're just beginning this you will find okay because this is just an introduction that will keep them positive all right so that's what the answers will be now the other aspect that i want to mention before i get into these examples that so for instance within here so as you're doing this when you have the square so for example it's the square root of a number it doesn't always give you a positive answer back there's it could be either positive or negative so what i mean by that is the following so if you have so let me just clear this up here so i'll delete that and then i'll delete that we're mostly used okay or we're mostly used to if we take the square root okay to give a positive answer and that is again because 3 squared is going to give us 9. but the square root of 9 can also be negative 3 again because negative 3 squared which is equal to negative 3 times negative 3 is going to be equal to 9 as well so what does that tell us well it tells us that if you're going to take the square root so typically what you might see as an answer if you're going to be writing you might see somebody write it's either positive or negative that will depend of course on maybe the application that you're working so on the problem sometimes negative answers will not make any sense so you only have a positive answer but if it's just for mathematical reasons you'll just put that the square root of 9 is plus or minus 3 because both of them actually work and the same thing is true if you had the square root of 4 it would be plus or minus 2 because negative 2 squared now again it's squared all in brackets is going to give you 4 back now this is not true necessarily when you have okay so these roots okay so when you have this root right here so this was for squared so when your root so this is n and this becomes your you're taking the nth root when it is even so for example when it is 2 when it is 4 when it is 6 8 and so on then the answer so your answer is going to be either positive or negative all right so that's what you will have and the reason is because okay when you have them both as pairs okay and you multiply let's say negative two times negative two that always gives you a positive back so as long as your roots are even then the answer can be either positive or negative if your root okay is odd so if your root is odd now one doesn't really make any many any sense okay so the first root of a number because it just gives you the number back so i will start with three so three five seven and so on in this case if you're radicand okay so if your radicand is positive then the answer must be positive right so that's what you will have if your radicand is negative then your answer will be negative okay so that's what you will have within there and i'll show you some examples as well with that all right so here okay if we're going through these questions so let us take a look and see what the answers are and i'll try to explain these okay just by taking out our calculator and then seeing through because these ones are a little bit larger okay so how would we do that so here's the fifth root of 243 all right so you can input this in and this is good for you in terms of trying to practice this out so this is the fifth root of 243 equals to three okay so the answer here is three now notice because five is odd so the answer can only be positive we cannot say that the answer is negative because if you took negative three and you took it to the fifth power so if you took so negative 3 and you brought it to the fifth power notice that the answer would have been negative 43 all right so to give you a contrast if you had a question and it was something like this and you notice that ah this is now negative then the answer here would have been negative three all right so that is for odd only okay so only for odd okay so for this second example so if you have this notice that here we have an even root so we're going to take the sixth root of this number now if you do that you can check it for yourself on a calculator or not okay this is going to give you a 7. now because this is even your answer is going to be either positive or negative so you can ask your teachers how they want this displayed if you're doing it in school or something like that but if you would take so here if you take negative seven and you would raise it to the power of six you would notice that you're going to get exactly the radicand back and if you raised this okay so seven to the sixth you're gonna get exactly the same thing back so that's why the answer can be both either positive or negative or it depends on the application sometimes negatives do not make sense the next one that you have in here so notice that the root is this is fourth root of 6.5536 so this is interesting because now it's a decimal now for decimals it doesn't matter you do them exactly the same way and you can check what the answer here would have been all right so if i take the fourth root now first of all because it's even it can be both positive or negative so the answer can be both positive or negative so 1.6 that's what we have there now this one right here and let me change it instead of having it positive i'll put a negative in there so if it's negative all right so what is the third root of 10.648 and this one the answer is going to be negative all right so that's going to be negative and you can check that for yourself so if you put this and you put let's say negative ten point six four eight all right equals two and it gives you two 2.2 all right which is negative all right so that gives you kind of an introduction to radicals all right and roots how to find them hopefully what they mean you do have to do a few examples what i will leave you off with is that you cannot especially as you're starting off now later on if you get into high school it is possible but those are going to be something called imaginary numbers but when you're starting okay you can ignore it when your n so when your n is even you cannot have a negative number underneath there okay if you want real number answers so keep that in mind um so for real number answers okay you cannot have these negatives within okay or you would just leave it as is so for instance if you took say the fourth root of negative and just just arbitrarily write this all right something like that and then that negative you cannot get a positive answer out of that so that's what i wanted to leave you with at the end all right okay so thank you for watching that finishes this video and we'll see you in the next video on these radicals bye everybody