Radical Expressions - Video Tutorials & Practice Problems

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1

concept

Square Roots

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5m

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Everyone early in the course, when we studied exponents, we saw how to square a number and we saw something like four squared was equal to 16. But now what's gonna happen in problems is they'll give you the right side of the equation like 16. And they're gonna ask you for the left side, they're gonna ask you what number when I multiply it by itself gets me to 16. And to answer this question, we're gonna talk about square roots. Now you've probably seen square roots at some point in math classes before, but we're gonna go over it again because there's a few things that you should know. Let's go ahead and take a look. So basically, the idea is that squares and square roots are like opposites of each other. The reverse of squaring a number is taking the square root. So for example, if I were asked, what are the square roots of nine, I have to think of a number when I multiply it by itself, it gets me to nine. So let's try that. Is it gonna be one? Well, no, because one multiply by itself is one. What about two? Now that just gets me four. What about three, three? If I multiply it by itself, you know, square it over here, I get to nine. But is that the only number that works for? Well, actually, no, because remember that negative three, if I square negative three, the negative sign cancels and I also just get to nine. So in other words, there's two numbers that when I multiply them by themselves, they get me to nine. And what that means is that nine has two square roots three and negative three. This actually always works for positive real numbers. There always have two roots. There is a positive roots like the three and textbooks sometimes call that the principal roots, but there's also the negative roots, the negative three. All right. So basically, if I start at nine and I wanna go backwards and take the square roots, there's two possible solutions I have three and negative three. So how do we write that? Well, we use this little radical symbol over here, this little um this little symbol. And so if I go backwards for nine, I get to three or negative three. But notice that there's a problem here. So if there's two possible answers for the square root of nine, how do I know which one I'm talking about? Am I talking about three or negative three? Because sometimes in problems you'll just see a square root like this. How do you know which one? It's talking about basically, it comes down to the way that you write the notation. So what we do here is the radical symbol when it's written by itself, that means it's talking about the positive root. So if you just see radical nine by itself, it's just talking about the positive root of three and to talk about the negative root, you have to stick a minus sign in front of that radical symbol. That means that now you're talking about the ne the the negative roots, which is the negative three. All right. So it's super important that you do that. Um Because what I learned when I was studying this stuff is that if you just have radical nine, you could sort of just write plus or minus three, but you can't do that. This is incorrect. And if you try to do this, you actually write this on a homework or something like that, you may get the wrong answer, right. So just be very, very careful. The notation is very important here. All right. And then what you also see sometimes is that if you want to talk about both of these at the same time, you'll see a little plus or minus in front of the radical. That just means that you're talking about plus and minus three. So both of them at the same time. So this is a little bit more efficient that way. All right. So that's all there is to it. So let's just actually go ahead and take a look at our first two problems here. If I want to evaluate this radical, I have radical 36. So in the words, I need to take the square to 36 and I need a number that multiplies by itself to get me 36. So let's just try. One squared is not gonna be that because that's just 12 squared is 43 squared is 94 squared, four times four is 16. So I have to keep going. I got five squared which is 25. That's still not it. And what about six squared? Well, six squared is equal to 36. So that means all of these are wrong answers. But this one is the right one. I have six when I multiply by itself gets me 36. So which one is? But that also means that negative 36 I'm sorry, negative six also gets me to 36. So what's the answer here? Is that the positive, the negative? Remember the radicals by itself. So this actually means it's just talking about six and it's not talking about both of them or the negative one. All right. So it's very important. What about the second one here? Now we see a negative that's in front of the radical symbol. That means it's talking about the negative roots of 36. So this answer is negative six over here. All right. So that's how to do those kinds of problems pretty straightforward. Let's take a look at this last one over here, which is now I have a negative, it's inside of the radical over here. All right. And to do this, we're actually going to talk about what happens when you get negatives inside of radicals. Because basically what that means is that you need to fi find a number that when you multiply it by itself gets you negative 36 or you know, in this case negative nine. Can I do that? Well, here what happens is if I try to do three, remember three squared is not negative nine, it's just positive nine. So that's not gonna work. And what about negative three? That's also not gonna work because if I took negative three and squared it, I, you know, uh I just got a positive nine. So in other words, that's not gonna work either because that just equals nine. So how do I take the square root of a negative number? It turns out you just can't do it, you can't do it because no matter what number you pick when you multiply it by itself, the negative just cancels out. Um And so what happens is all you need to know for right now is that whenever you see a negative that's inside of a radical, you just need to know that it's imaginary and we'll cover this later on. But that's all you need to know for now. So here's a good sort of like a memory tool to use when we saw negatives that were outside of radicals, that was perfectly fine. And that was OK. So for example, we saw negative outside of radical nine or 36 that's perfectly fine. But if you see a negative inside, that means that it's imaginary. So outside is OK. But inside is imaginary. All right. So negative radical 36 over here perfectly fine. But the radical of negative 36 that's imaginary. All right. That's all you need to know for now. Anyway, folks, so that's all there is to it. Let me know if you have any questions. Thanks for watching.

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Problem

Problem

Evaluate the radical.

$-\sqrt{\frac14}$

A

$\frac12$

B

$-\frac12$

C

$-\frac{1}{16}$

D

No real solution (Imaginary)

3

Problem

Problem

Evaluate the radical.

$\sqrt{\left(-5\right)^2}$

A

$2.23$

B

$5$

C

$-5$

D

No real solution (Imaginary)

4

concept

Nth Roots

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6m

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Welcome back everyone. So we saw recently that squares and square roots were like opposites of each other. What I'm gonna show you in this video is that squaring isn't the only exponent that we can do. We can also do numbers to the third power or the fourth power or so on and so forth. What I want to do here is just talk more generally about roots and I'm gonna show you that roots really just fall into two types of categories and I'm gonna show you the differences between these categories. Now, let's get started. I'm gonna actually get back this information later on in the video. I'm just gonna go ahead and get to the numbers because I think it'll be super clear here when we did square roots, we said that two squared was equal to four and negative two squared was also equal to four. So both of these numbers were square roots of four. And that means that if you go backwards from four, if you undo that by taking the square roots, you get two and you should get negative two. So does this happen for other exponents though? So does this happen if I take two to the third power. Well, let's just take a look here. What's two to the third power? It's two times two, which is 44 times two is eight. Negative two. What happens is the negatives cancel for the first two terms. But then you have another factor of negative two and this turns into negative eight. So here's the difference when I took two and negative two and I squared them, I got just the same number of four. Whereas when I cube two and negative two, I get different numbers eight and negative eight. So just as the square root was the opposite of the square, then we can do cube roots to take the opposite of the cube. And what we see here is that the cube root of eight is not both of these numbers, you don't get two numbers because it only just gets us back to two and it's not negative two, negative two gave us negative eight when we cubed it. So the Q root of eight is just two and the cube root of negative eight, if I work backwards from this number just gets me to negative two. All right. And that's the main difference here. Whereas for square roots, we always saw two roots, there was a positive and a negative, the two and the negative two and they were the same. Whereas for H roots, what happens is we always always have, always have one roots roots are always actually the same sign as the Radicans. So that's also what we saw. The two is the same sign as the eight, the negative two is the same sign as the negative eight. And furthermore, what we also saw is that when we have negatives inside of Radicans, the answers to those were imaginary nothing when squared gave us a negative number. So the answers were imaginary. Whereas here for cube roots, what happens is if you have negatives inside the Ratican, that's perfectly fine. Your answer actually just turns out to be negative negative two. If you multiply it by itself three times gets you to negative eight. So negative numbers inside of cubes are perfectly fine. So here's the whole idea more generally, if you take a number and you raise it to the nth power, the opposite of that is taking the nth root. So in other words, if I have a number like a and I raise it to the N power like the third power, fourth power or something like that, then the opposite of that is if I take the answer and I take the nth root of that, I should just get back to my original A. That's sort of more generally what happens. This num this letter N here is called the index and it's written at the top left of the radical. For example, we saw the three over here, but you also might see a five or a seven or something like that. Um The only thing you need to know though is that for square roots, there's kind of like an invisible two here. Um So the square roots, the N is equal to two, but it just never gets written for some reason. All right. And furthermore, what we saw here is that square roots and cube roots are really just examples of where you have even versus odd indexes. So everything that we talked about for square roots, the two roots and the imaginary stuff like that, all that stuff applies for when you have even indexes like fourth roots, sixth route, stuff like that. And everything that we talked about over here for cube roots also applies when you see fifth roots and seventh roots stuff like that. So what I like to do in my examples is I look at the number inside the radical, look at the index and I just go over here and use these rules, but that's all there is. So let's go ahead and take a look at some examples. So we're gonna take a look at the following nth roots and evaluate them or indicate if the answer is imaginary. Let's get started here with the roots of fourth route of 81. What I like to do is actually always look at the number inside um and figure out if it's positive or negative. So negative one or sorry, 81 is positive And so what that means is that now we look at the index. So if I have a positive number and then I look at the index, what that tells me is that I'm gonna look at these two rules over here. I should have two roots, one is negative and one is positive. So what is the fourth route of 81? Well, rather than having to sit here and calculate a bunch of stuff. But I've actually sort of done for you is I've come up with a list or a table of perfect powers like perfect squares or cubes or even other powers that are gonna be really helpful for you to uh you know, to memorize, you don't actually have to memorize them. You could always just recalculate them if you need to. But let's go here. So I'm just gonna look for 81 inside of this list. I see 81 is nine squared, but I'm not looking for something squared. I'm looking for something to the fourth power. So if I keep looking over here, what I see is that three to the fourth power is 81. So the opposite of that is that the fourth root of 81 should just give me three. Now, remember what happens is this radical symbol because it's positive, just means that they're talking about the positive roots. So the answer is just three, right? Because Greece to the fourth power is 81. Let's take a look at the second one here here. What we have is I have negative 32 and I have the fifth root of that. So take a look at the number first, it's negative. So what does that mean? I look at my two, I look at my index and it's an odd index negative inside of a radical for odd indexes. The answer is just going to be negative. All right. So I look at my list over here, I'm gonna try to find what thing when multiplied by. So five times gets me to 32. And you'll see here that two to the fifth power is 32. So in other words, what happens is negative two to the fifth power. If you can multiply this out just gives you negative 32. Therefore the fifth root of 34 is just equal to negative two. All right. So look at the number first, figure out if it's positive or negative, then look at the index and that'll tell you which rules to use. All right, let's look at the last uh the third one over here here. What I have is I have a negative number. All right. So then I look at the index. So is it even or odd? Well, I have a negative number with an even index. So that means I look at this rule over here. So a negative inside means that the answer is imaginary. So this just equals an imaginary number. And that's all you need to know for. Now, that's the answer. Take a look at the last one in the last one here, I have negative five to the seventh power. So this is not a number, but it's actually just something that's gonna be raised to a power of seven. And then I have to take the seventh route of that. Now, rather than having to sit here and calculate what negative five to the seventh power is. If, if you uh I'm actually gonna show you a really cool shortcut for this. Basically, what happens is I'm gonna take a number and I'm gonna raise it to the seventh power and then I'm gonna take the seventh route of that. So basically, those are just opposites of each other. If you ever have a term in a radical where the exponents equals the index. In other words, I have a seven here as an exponent and that's the same thing as the seventh uh roots. And they cancel out. I take number race to the seventh power and then 1/7 root. It, it's basically like I'm just canceling it itself out and then all you're left with is just whatever was inside of the radic hand. So your answer here is just negative five. All right, folks. Thanks for watching. That's it for this one.