0. Fundamental Concepts of Algebra
Radical Expressions
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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 going to show you in this video is that squaring isn't the only exponent that we can do. We can also do numbers to the 3rd power or the 4th power or so on and so forth. What I wanna do here is just talk more generally about roots, and I'm gonna show you that roots really just fall into 2 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 to 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 2 squared was equal to 4 and negative two squared was also equal to 4. So both of these numbers were square roots of 4. And that means that if you go backwards from 4, if you undo that by taking the square roots, you get 2, and you should get negative 2. So does this happen for other exponents, though? So does this happen if I take 2 to the 3rd power? Well, let's just take a look here. What's 2 to the 3rd power? It's 2 times 2, which is 4. 4 times 2 is 8. Negative 2, what happens is the negatives cancel for the first two terms, but then you have another factor of negative 2, and this turns into negative 8. So here's the difference. When I took 2 and negative 2 and I squared them, I got just the same number of 4. Whereas, when I cube 2 and negative 2, I get different numbers, 8 and negative 8. 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 8 is not both of these numbers. You don't get two numbers because it only just gets us back to 2 and not negative 2. Negative 2 gave us negative 8 when we cubed it. So the cube root of 8 is just 2, and the cube root of negative 8, if I work backwards from this number, just gets me to negative 2. Alright? And that's the main difference here. Whereas for square roots, we always saw 2 roots. There was a positive and a negative, the 2 and the negative 2, and they were the same. Whereas for cube roots, what happens is we always always have always have one root. Roots are always actually the same sign as the radicands. That's all also what we saw. The 2 is the same sign as the 8. The negative 2 is the same sign as the negative 8. And furthermore, what we also saw saw is that when we have negatives inside of radicands, 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 radicand, that's perfectly fine. Your answer actually just turns out to be negative. Negative 2, if you multiply it by itself 3 times, gets you to negative 8. 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 3rd power, 4th power, 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 numb this letter n here is called the index, and it's written at the top left of the radical. For example, we saw the 3 over here, but you also might see a 5 or a 7 or something like that. The only thing you need to know though is that for square roots, there's kind of like an invisible 2 here. So the square roots, the n is equal to 2, but it just never gets written for some reason. Alright? 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 2 roots and the imaginary stuff like that, all that stuff applies for when you have even indexes like 4th roots, 6th roots, stuff like that. And everything that we talked about over here for cube roots also applies when you see 5th roots and 7th roots and stuff like that. So what I like to do in my examples is I do 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, 4th root of 81. So what I like to do is actually always look at the number inside, and figure out if it's positive or negative. So negative 1 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 2 roots. 1 is negative and 1 is positive. So what is the 4th root of 81? Well, rather than having to sit here and calculate a bunch of stuff, what 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, 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 get started here. So I'm just gonna look for 81 inside of this list. I see 81 is 9 squared, but I'm not looking for something squared. I'm looking for something to the 4th power. So if I keep looking over here, what I see is that 3 to the 4th power is 81. So the opposite of that is that the 4th root of 81 should just give me 3. 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 3. Alright? Because 3 to the 4th 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 5th root of that. So take a look at the number first. It's negative. So what does that mean? I look at my 2 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. Alright. So I look at my list over here. I'm gonna try to find what thing would multiply by itself 5 times gets me to 32, and you'll see here that 2 to the 5th power is 32. So in other words, what happens is negative 2 to the 5th power, if you can multiply this out, just gives you negative 32. Therefore, the 5th root of 34 is just equal to negative 2. Alright? 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. Alright. Let's look at the last the third one over here. Here, what I have is I have a negative number. Alright. 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. Let's take a look at the last one. In the last one here, I have negative 5 to the 7th power. So this is not a number, but it's actually just something that's gonna be raised to a power of 7, and then I have to take the 7th root of that. Now rather than having to sit here and calculate what negative 5 to the 7th power is, if if you, 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 7th power, and then I'm gonna take the 7th root 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, so in other words, I have a 7 here as an exponent, and that's the same thing as the 7th roots, then they cancel out. If I take number raised to the 7th power, and then I 7th 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 radicand. So your answer here is just negative 5. Alright, folks. Thanks for watching. That's it for this one.
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