College Algebra | Section 2 | Introduction to Radicals | Victor Rodriguez | Skillshare

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College Algebra | Section 2 | Introduction to Radicals

teacher avatar Victor Rodriguez, Teaching Math and Science

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Lessons in This Class

7 Lessons (55m)
    • 1. Introductions & Definitions

    • 2. Working with Variables in Radicals

    • 3. Simplifying Radicals

    • 4. Adding/Subtracting Radicals

    • 5. Multiplying Radicals

    • 6. Dividing Radicals

    • 7. Intro to radicals Review

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About This Class

This course will build on the understanding of polynomials by introducing you to the concept of radicals. This course will introduce you to what a radical is and how to handle them within the greater polynomial operations. 

Prerequisites: It is advised that you first take my course "College Algebra | Section 1 | Introduction to Polynomials" if you need a refresher on polynomial or have not yet been introduced to the concept of polynomials. 

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Victor Rodriguez

Teaching Math and Science


Hello, I'm Victor. I am an astrobiology researcher and a math and science enthusiast. I love sharing my knowledge with the world. 

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1. Introductions & Definitions : Hello, students. This is Theo. Introduction to radicals. Uh, course, my name is Victor. I'll be your instructor. And first, I'd like to let you know that there are some course documents that you can actually download in. Follow step by, step with each video. And so make sure you download those that where you have some extra practice while you get some instruction. The this is course number two. There's, of course, number one. And so on that you can also check out, uh, let me know if you have any questions. You can email me. My email is in the course description and let's get started. So this is, ah, introduction to radicals. Um, first, I'd like to go ahead and do some some definitions. Now, the parts of a, um radical is the Radic cam that goes inside the, uh, radical bracket. And on the and on the outside of the bracket is the index. I'm gonna do some definitions. Okay, So I find the terms such that when you multiply by itself equals the radical the radical. Now you can kind of put now you can actually add to that definition. Find the terms such that when you multiplied by itself the index number of times it equals the radical and and that's, you know, nutshell What a radical is now, if the index is not given, it's assumed that it's too. Now we're gonna do some or definitions here. Now we're gonna look at first. We're gonna look at rational numbers, not for a number to be rational. For number to be rational, it must either be able to be written as a fraction it or terminating decimal or be a repeating decimal. Now, examples of this, if we go over here, is 1/3 is a rational number. A terminating decimal would be 0.25 That would be also irrational number or, in the case of 1/3 point 333 forever would be a rational number as well. Now we're gonna talk about irrational numbers. An irrational number is a number that does not terminate. It doesn't repeat, and it cannot be written as a fraction Now. There's actually a pretty famous example of a irrational number in that's pie. No matter what you do, there were how many decimal places you go out, it doesn't end, and it doesn't repeat its 3.14 blah, blah, blah blah and keeps going, Um, and it doesn't repeat its not like 3.14444 forever. It's an irrational number that what I've created here is a table of rational and irrational numbers. A Z. You can see her, all the rational ones you have 1/3 which is could be written as a fraction. And, of course, this is three and 1/3 which it's a repeating decimal. Keep going now. The repeating decimal doesn't have to be, um, a single number like 3333 It could also be 4545 continuous forever, and 12.5 is a terminating number. Now, of course, as you can see with all of these, there's no um hence the term There's no rational pattern to any of these decimals. They just keep going on forever, with no repeating pattern to them. Now we're now, we're gonna talk about perfect numbers, and as you can see, I'm kind of building up to something here with this. Is that in this situation, if n equals a, then it's a perfect number as well. Now I'm gonna show you this numerically if you have 25 with n equals two which normally you don't put to you just assume, But I'll put it on there for right now. Now, of course. 25 the square root of 25 5 Well, let's break this down. If five equals X than 25 equals x squared. And under this scenario, this equals a In this two over here equals n So, as you can see here, n equals a because then is two in a two. Because right in here you can take this and write it as five squared. So this is N This is a This is X. So perfect numbers any number that could be routed to give you a either a square or a cubed or whatever the end radical is. So that was your introduction into radicals. Next video. We're actually going to go a little further and start adding variables into radicals. 2. Working with Variables in Radicals: Okay, now, Now, in this video, we're gonna talk about, uh, using radicals with variables X to the M with a root of end is a perfect number on Lee. If m divided by N is a whole number now, what I'll do is I'll actually, uh, show exactly what What I mean by this. They're just as when you divide exponents, you subtract the exponents, you the exponents. So if you have, um, X to the 10 divided by X to the five, it would equal X to the fifth power. Well, in this case, when you read it, when you're doing radicals, you actually subtract. So in this case, X to the nine with a Q group equals X to the three. And as you can see here, it's a whole number. And so this is a perfect number. And the reason you can you can see this is because of you. Take nine. Divided by three equals three. So this so x 2 to 9 is a perfect number. On the other hand, this is not a perfect number. Extra nine. Um, with the fourth root does not equal a perfect number, because if you take nine divided by four, you'll get 2.25 Therefore, this is not a perfect number. So the in this case, what the way you would write it is x nine over four. And this is how you This is how you would write this number in a non radical form. So that's it for ah, perfect numbers. Next, we're going to start talking about simplifying radical numbers. 3. Simplifying Radicals: how everyone in this video I'm gonna be talking about simplifying radicals. Now, this is a pretty long definition of exactly what we'll be doing when we simplify radical right rat right Radic, and as a product using the highest number, that is perfect, a perfect factor that divides into it. So let's do this in practice, you know, let's take radical 18. Now we're assuming, because there's nothing there that this is a two for the, um, Index. Now, what we can do is we can actually split this up and and try to find the, um, the highest perfect number. Now, the highest perfect number that goes into 18 is nine. So we can actually divide this into nine times two, which is 18 then. Now that we've done this, we can take that in the square root of nine is three and then we leave the radical to alone . So what you do is you try to find the highest perfect number removed that from the situation or divide, divide the ah Radic and by the perfect number. And now you can separate that, and then you can take this and now you can move it out into the numbers. And now you have three radical too, this time radical three or 250 to the Cube group. No, this is where it's good to memorize your perfect numbers, your perfect squares and perfect cubes. It makes things a lot easier. So if you could think of a perfect cube, one perfect cube is 1 25 So you can then take this and you can write it as such. So now 1 25 times two of the cube root of 11 25 times two. Now we can then take this, which the cube root of 1 25 is five and then radical her cube root to. That's how you saw that. It is very important that you are able to memorize your perfect cubes that will make perfect squares and perfect cubes because that will make your life a lot easier when you deal with radicals. Now, we're going to get a little more complicated when it comes to simplification. Those were single numbers, never gonna throw in variables and in the way. So what we're going to do in this situation, of course, that the indexes to what we're gonna do is we're gonna take things piece by piece here in under it, we're going to do another radical sign that we've been start simplifying and taking things apart. Now, 24 the closest thing that we can get here Ah, perfect is four. So we can take four times six is 24. So we take four time six. You know, that takes care of that when you're dealing with exponents as a radical can What? As you remember, with perfect numbers, whatever the exponents is divided by the index must be a whole number. So first we have to figure out what's a whole number for this. And in this case, because this is too. It's pretty easy. We just have to find an even number. Now, as you remember, with multiplying exponents, this is actually pretty easy. What you can do is the in this case, you find the highest even number, which is four. So what we do is X to the fourth, and then you take X to the one because exited fourth and exited. One is times exited. One is exit fucked to the fifth, then with this one. This one's easy because this one's an even number. So we can We can leave that one alone and this one's an even number so we can leave that one alone. Now we can go through and work our whole numbers out. So first, what we'll do is we'll take out our whole numbers, which the square root of four is, too. That deals with that. Now this one isn't a perfect number. So we have to leave that alone. And then this will be a perfect number because it's even so you can take. And of course, the way you do This is X squared is you take four divided by the index, which is too. So it would be X squared. Then this one we leave alone and then we go to this one. Why? To the six? Why six is a is a even number So we can divide six divided by two, which is the index and I'll give you three. So it's why cubed and then Z to the fourth if you take four divided by two and it's two squared So Z squared. And then now the ones that are left Yeah, we leave in the radical we take in as erratic and six x So the answer here is to X squared y cubed Z squared radical six X And as you can see there, what we did we started taking apart, um, each variable in each coefficient to give us the number. Now I'm gonna do another one because this can get pretty confusing. So it's good to have a good amount of practice with this now, in this example, the Radic An is the same, but I've changed the index, so it completely changes everything. Now we're looking for the the perfect cube to go into 24. So let's draw out or radical symbol. Now, in this case, the perfect number that is a cube in this case would be eight. Because if you take 24 and 82 times two is four, two times four is eight. So that's a perfect cube in this case. So you take eight is the highest perfect cube, times three. And now that deals with 24 now, In this case, now we're trying to find the highest multiple of three are the the highest exponents. There's a multiple of three. In this case. It would be three, three times one is three. So you take X cubed. And then now you have two left. So three times two is five. So that deal that takes care of that and then you have white is six. Now, in this case, six divided by three is too. So it works perfectly. So we can Actually, it's a perfect number. So we can actually leave that alone. And then finally with a four, the highest, uh, multiple again is three. So it's three for Z Cube and then now we take the rest of it, which is just a Z. Now this is ready to be simplified. First, let's do the whole numbers. Eight. The cube root of eight is to we have to leave three alone and then the cube root of X to the third is just x so that deals with that X squared cannot be, uh, you can't take the cube root of that. So now why? To the six if you take six, divided by three is too so you can take why squared and that one's done and NZ cubed, Divided by r. The cube root of Z two cube. If you take three divided by three is one. So you can just take Z that deals with that and in this city is, um, can't be messed with. So now we deal with the radicals with radical can We moved a three down the X squared down and the Z down. And now, in this case, the answer is to X y squared Z with the cube root of three X squared Z. And you can see how just changing the index can change completely. Change a simplification of this. So that's the video on simplifying radicals. Um, next, we're going to actually start doing operations with radicals. 4. Adding/Subtracting Radicals: so operation with radicals. Now there's some goals in mind whenever you're doing operations with radicals, and I'm going to go through those goals right now. First the home, Oh, answers must be simplified at the end. That's the goal. That's one of the goals I kind of scrunched there at the end. But, uh, no radicals in the denominator all like terms must be combined. The final goal is to reduce all coefficients, and this will make a lot more sense when I start going through different problems. No, no. In this video, I'm just going to go through addition and subtraction. Now, when you're adding and subtracting, you have a couple goals in mind. Now, the first step is each radical should be simplified first, and then the final step is you should combine all like terms in this for addition. There should work just fine. Now that we've gone through the definitions, I'm going to go do two examples to kind of give you exactly what I'm talking about, because ah, lot of times it just doesn't really make sense until you actually see it in practice. Now, as you can see here, you have ah radical 32 x squared minus four x radical 18 x plus radical 36 x squared now. So first, what we want to do is simplify everything. So we're gonna go from left to right in each term and simplify everything. So because all of these air square So this is ah, the indexes to here. So the closest perfect square is is 16 16 times two and an x squared. We can leave that alone and in minus. We re write these, leave that there and then we have 18 X, which the highest perfect square is nine, nine times two and an ex. We have to leave alone because it's not a perfect number. And then plus, and in here is actually pretty easy because 36 is a perfect square and an X squared is a perfect square so we can leave that alone. Now that we've separated all the the perfect numbers out of the radic ans Now we can simplify them all. And then let's start with the whole numbers. The square root of 16 is for, and then the square root of X squared is X. And then what's left is to and then negative for now. In this case, it gets a little complicated now that we have this here again. The best ways toe methodically do things. Um, four X and then we'll continue on the square root of nine is three. So will leave this here so it's four times x times three and then the two X has to be left in because it's not a perfect square. And in plus, finally, we This one is easy. It's we can take out 66 times 6 36 and an x square root of X squared is X. Now, the final step is that we can take this and we can, uh, reduced coefficients and, um, combine like terms. Now, in this case we have we can leave this one alone for ex radical too. And then this one minus, we can take, uh, four times three is 12 x radical two X plus six X Now, in this case, we can actually, as as you know, from polynomial, when you're combining like terms in this case, you can treat a simplified radical as a variable. The thes terms cannot become combined any further because you have four x radical too. Well, you have 12 X which these could be combined. Except for this isn't two X, not a rat, A radical two x Not a radical, too. If this was a radical too, then we could combine these. And of course, this has no radical at all. Therefore, all of this right here is the final answer. Now I'm gonna do another one demonstration to kind of give you a little, uh, more practice. Now, in this case, we have the cube root of 16 in the Cube root of 54. Now, in this case, there's no variables, so makes little easier. But I'm doing this one to show you how you can combine like terms. So first, the Q brew of 16. It's not a perfect number, but we can get the Q group of eight times two. And then now, in this one 54 isn't a perfect cube. However, 27 goes into 54 in 27 is a perfect you. Now this is ready to be simplified. Now, First we can take out the eight, which gives you two with a cube root of two and then you take 27 which comes out to three with a Q group of to now. In this case, we can combine like terms because you have the cube root of three, the cube root of three. You have a coefficient, which coefficients can be combined, and in this case there's no variables that are different. Therefore, this these are like terms, and we can combine them. So now the way this works is you have you treat him like variables. It's it would be like doing two X plus three x. This would be five. So in this case, two plus three is five, and then you just bring in the radical, too, or the cube root of two. Don't be tempted to try to do any arithmetic with these. You have to treat him like variables. Otherwise, when you're adding them together and combining like terms, you end up, um, messing up and changing the values of the terms that you're trying to combine. This is my so this is the video on adding and subtracting radicals. My next video will be multiplication of radicals 5. Multiplying Radicals: however, one This is the video on multiplying radicals in this step before for this, we're gonna be talking about same index. Um, what you do is you multiply and then you simplify pretty much similar, um, with the coefficients on the outside and, well, let's do some practice with this. Now, in this problem, we have, um radical eight extra fifth times three radical four x to the, uh um four x q r squared. So what first thing we want to do is we want to simplify. And of course, if you, uh, take the rules from simplification, we can rewrite this as four times two because for is a perfect square. And then we can subtract the highest, even number, which is four. And then we put the so X fort off X to the fourth times X and then we can take this. Leave the coefficient alone. In this case, this is a perfect square, so we can leave that alone. And of course, X squared is a perfect square, so we can leave that alone. Now we can simplify them. Square root of four is too square root of X to the fourth is X squared and then the two when the X can be left alone. Now it's in there. We have that. And then we can take the three. And in the four x times that the square root of force to this group square root of X squared is X. So now now we can multiply these there is a mono mealtime in times of my no meal. So it's pretty ST ST Forward, um, to make things simpler, simple. First we can take this and turn it into six. And then now we now down here, we can actually do the math. So the coefficients, six times to his 12 that takes care of the coefficients. And then we have X squared Times X, which is X cubed so that Kate takes care of the exes. And then finally, this one can be left alone because there's no radical to multiply it with. So you have two X, and the answer is 12 x cubed radical two x. So the first thing we want to do now, of course, this is a polynomial times a polynomial. So we do for you. You take now when you're when you're multiplying, uh, to radicals like this. What you want to do is you multiply that cowfish is you multiply what's on the outside and then multiply What's inside? So this would be a time see radical B times D. So let's apply this to this situation we have. You can imagine there was a one right here. So four times one is for and in five times 10 is 50. So this will be radical 50 and then So that takes care of these. And then we have four radical five times negative too, which equals eight radical five. Then we go on to the positive too. Ah, two times radical 10 equals to radical 10 and then finally neck two times. Negative too is negative. For now we can take this and we can combine like terms and ah, simplify as well. Now, in this case, we're gonna work on simplifying. So we take four and in here you can take 24. 25 has a perfect square in 25 times two is 50 and then minus eight. And of course, there's nothing you can do with radical five to simplify it. Plus to radical 10 minus. For now, this is now. This could be simplified and of course, is a lot of steps, but I like to do things methodically. That way you can see the the resulting, um methodology. So now you can take the 25 out the 25 square or the square root of 25 5 so in in five times , so you can start with the four times 25 radical 25 5 radical, too minus Hey, radical five plus to radical 10 minus four. And then we can then Teoh complete the simplification. Five times, four times five is 20 radical, too minus eight radical five plus two. Radical 10 minus four. And that's your answer, and that's how you multiply radicals. 6. Dividing Radicals: Hello. Everyone in this video we're gonna talk about multiple are dividing radicals. No. First I'm gonna have an illustration to show you how the relationship of fractions are within a radical. If you have radical a over B, this is the equivalent of radical a over radical Be these two expressions are exactly the same. And this intuition could be used when you're doing division off radicals. So here are the steps. When you're dividing radicals, first you want to simplify and then cancel. What I mean is you simplify all the radicals in, and if there's any, oh, radicals that that could be canceled out or anything else I mean, canceled out. You cancel it out. So if there's, um in X squared over X squared, that would be a one, you can cancel that out. Now, Rationalizing the denominator means that in the end, you do not have a radical in the denominator, and I will explain that more later now related to this, the denominator should be a perfect number. And finally, what you do is you reduce the coefficients. You have radical 6/7 now first, what we do is we separate this so you have radical six over radical seven, then neither one of these air perfect squares. So we can't, uh, simplify them. However, now comes a step off rationalizing the denominator. Now, the way rationalizes denominator is by squaring, you can multiply this by itself. What you do is you take the radical and you multiply it by itself. And that will give like we'll give you Ah, the erratic and so radical seven times radical seven. He's seven. However, you have to do it to both sides because with any fraction, the only way you can properly do this is by multiplying it by seven over radical seven over radical seven, and that will give you the same exact fraction, but in a different form. Six times seven is 42. So you would get radical 42 over seven. Now we can see if we can simplify. And of course, there's no square root. Uh, there's no perfect square that goes into 42. So in this case, we're done. And as you can see here, we we try to simplify. We couldn't. So then we took this. We multiplied it by 70 the denominator. Seven hours over seven to take the radical out of the denominator. You do that and you get 42 radical 42/7. Now, First, we can simplify this radical five over radical 16 times three because 16 16 times three is 48. Now we can then simplify this in its radical five over four Radical three. Now it's now it gets a little messy. Now, what we do is we. Now we have to rationalise the denominator. Take this radical out of the denominator in the way you do. This is again is you multiply radical three over radical three. And then, of course, what you do is radical. Five times radical three is radical, 15 over and in four, four times one of you imagine a one there is for times in a radical three times radical three is radical nine, which is three, which then you can simplify further to 15 over 12. Now, in this case, there's no coefficients to cancel out or anything. So we're done with that one. That is the answer. Now we're gonna do dividing radicals with two terms in the denominator. Now it gets a little now getting a little trickier And what you must remember is an extra rule. Is you multiplied by the country? Get now. Explain exactly what that means in a minute. Now, for this example, let's take radical three over radical five X plus four if we multiply. If we if we try to rationalize this by multiplying it by five x plus four will actually get a messy result and it will not eliminate the radical. So instead we multiplied by the congregate. Now this is the conjured it right here. If we take a plus B, the conjure git is a minus. B, this is the counter get of this. Now what we do is we multiply this by the congregate, which in this case is radical five X minus four over radical five X minus four. And now we can multiply this out and you'll see how this can get a little messy. Radical three times radical five X is radical 15 x and then radical three times negative four is negative for radical three and then this one here and then for this one Radical five x times radical five x is radical 25 x and then radical five x times negative. Four is native four radical five x and then four times radical five x is four times five x and actually this one squared. Miss that up four times four x and then finally 44 times negative four is negative. 16. So now the now we can combine like terms and we'll go down here for that. We'll start with the the top here. Now we can't combine these So it's 15 x minus four. Radical three. Now this is where the congregate comes in. 25 X squared can be simplified to five or can be Ah, simplified to five X because these air both perfect squares that takes care of that and then four. So we have negative four, uh, radical five x plus for radical five X. So this actually cancels out, and then we just bring the 16 the negative 16 over. So, as you see here, you have 15 x a radical 15 X minus four radical three over five X minus 16. Remove the radical out out of the denominator. And if you do this right, all the all the radicals should cancel out or simplify. If you don't do it right you'll notice here that something goes wrong and the radical doesn't cancel, and then you can kind of look and check to see if there's something that you missed. So this is how you do radical division with multiple terms in the denominator and that completes our radical division video for now, and the next video will be a review of the worksheets. 7. Intro to radicals Review: how everyone? This is the review for the introduction Radicals. Course I'm gonna do a, um, a problem from the simplification worksheet. And two from the additional subtraction multiplication sheet and then one from the division worksheet. So you can kind of tell you which numbers they are, and then you can follow along so we'll start off with this. Um, in this case, it's 16 because 16 is a is a perfect number that goes with to to their fourth of 16 times eight and to the eight. Because this is a divisible by four. We can leave that one alone. And then now we can simplify the cute are the, uh to do. The fourth is 16 so we can take to to the outside, and that eliminates that. And then we have into the eight which comes out to and squared, and then we can leave eight. And that's how you simplify that. So this is number six on the adding, subtracting, multiplying worksheet. Start off the first step. Of course it is simplifying, so this one can be simplified. We have a four in there, so let's do a negative three. Radical four times three and then this one plus three radical three. There's no simplification for that. And then plus three radical. In this case, we could do four times five. And now we can, uh, And now we can go ahead and right these out. We have negative three times. This is ah for square root of force to so you end up with negative six radical three and then plus three radical three and then plus six Radical five. And then these. Because these air like radicals, they are like terms. We can combine them. So negative six plus three is negative. Three radical three plus six radical five in that right there is the answer to number six. Can the adding subtracting multiplication? This is a multiplication problem in action, adding as well. So let's do this is number 20 on that sheet. First, I'm gonna start. There's no simplification here. There's no, uh, no simplification. Simplification here, here, so we can rule those out. Now we can foil. It's a negative. Negative. Radical, too. Now, that's different than radical. Negative to which is a, um, imaginary number or complex number, which, actually you can check out. I have a course just on complex numbers coming up, but that's a different issue entirely. So this is negative. Radical, too. Times radical 10. Well, of course, two times 10 is 20. So it would be negative, radical 20 and then multiply this negative times a negative is a positive. So in this case, you have plus for radical two times six is 12. And in this case now we can simplify it. Um, so this would be negative. Radical. Four times five plus four radical four times three. We can then simplify this further and we get negative to radical. Five plus eight radical three. And that's the furthest we can go with that. Okay, now this is number eight on the Dividing Radicals Worksheet. This is radical three plus three radical five over to radical eight. Now, the first thing we want to do is we want to take the radical out off the denominator. The way we do that is we multiply by two radical eight. - Okay , now chosen problem 10 on the dividing radicals. And I chose this one specifically because I wanted to come to get you are used to something that people tend to forget Is the idea of multiplying by the congregate. Normally we would multiply the top and bottom by whatever the radical is. But because this is a polynomial, we have to use the congregate. So I want to get you used to doing this. So what we do is we multiply this by the congregate, which is five minus radical, too over five minus radical, too. Multiply that and what you get is you do foil and, uh, you end up. Will you end up with this five radical five times negative. You have five negative radical 10 over by radical five or five times five is 25 and it five times negative. Radical to is five negative five radical too. And then multiply Radical two times five, You get five radical too. And then finally you get radical two times negative radical to which equals negative too. And then now you work this out and what you have is these to cancel out and what I'll do is I'll write all this down here. You end up with five radical five minus radical 10 over. Then you have 25 minus two. That's 23 now. You can't simplify this further, so That's ah, how you do that. And I wanted to make sure to remember that, um, the confidence. And this is true whether you have to polynomial if you have, let's say, well, the congregate of this would be five minus radical two plus radical eight minus seven. And that's how you get the congregate of this problem. You just flip all the signs. So that's how you do dividing radicals. This is a review, and hopefully this course has helped you out. Understand how radicals work in Check out my other courses on in the series of college algebra. Thank you.