Radical Expressions: Study with Video Lessons, Practice Problems & Examples

Understanding square roots involves recognizing that they are the inverse of squaring a number. For example, the square roots of 9 are 3 and -3, but the radical symbol (√) denotes only the positive root. In contrast, cube roots yield a single result, as seen with 2³ = 8 and (-2)³ = -8. The general rule is that even roots (like square roots) can yield imaginary results for negative radicands, while odd roots can produce negative results. This distinction is crucial for evaluating expressions involving roots and exponents.

Square Roots

Hey everyone. Early in the course when we studied exponents, we saw how to square a number and we saw something like 4 squared was equal to 16. But now what's going to happen in problems is they'll give you the right side of the equation, like 16, and they're going to ask you for the left side. They're going to ask you what number, when I multiply it by itself, gets me to 16. And to answer this question, we're going to talk about square roots. Now you've probably seen square roots at some point in math classes before, but we're going to go over it again because there are 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 9, I have to think of a number. When I multiply it by itself, it gets me to 9. So let's try that. Is it going to be 1? Well, no. Because 1 multiplied by itself is 1. What about 2? Now that just gets me 4. What about 3? 3, if I multiply it by itself, you know, square it over here, I get to 9. But is that the only number that works for? Well, actually, no. Because remember that negative 3, if I square negative 3, the negative sign cancels, and I also just get to 9. So in other words, there are two numbers that when I multiply them by themselves, they get me to 9. And what that means is that 9 has 2 square roots, 3 and negative 3. This actually always works for positive real numbers. They always have 2 roots. There is a positive root like the 3, and textbooks sometimes call that the principal roots, but there's also the negative roots, the negative 3. Alright? So, basically, if I start at 9 and I want to go backwards and take the square roots, there are 2 possible solutions. I have 3 and negative 3. So how do we write that? Well, we use this little radical symbol over here, this little, this little symbol. And so if I go backwards from 9, I get to 3 or negative 3. But notice how there's a problem here. So if there are 2 possible answers for the square root of 9

9 , how do I know which one I'm talking about? Am I talking about 3 or negative 3? Because sometimes in problems, you'll just see a square root like this 9 . 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 9 by itself, it's just talking about the positive root of 3. 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 negative roots, which is the negative 3. Alright? So it's super important that you do that, because what I learned when I was studying this stuff is that if you just have radical 9, you could sort of just write plus or minus 3, 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. Alright? 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 3. So both of them at the same time. So it's a little bit more efficient that way. Alright. 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 36 . So in other words, I need to take the square root of 36, and I need a number that multiplies by itself to get me 36. So let's just try. 1 squared is not going to be that because that's just 1. 2 squared is 4. 3 squared is 9. 4 squared 4 times 4 is 16. So I have to keep going. I got 5 squared, which is 25. That's still not it. And what about 6 squared? Well, 6 squared is equal to 36. So it means

Evaluate the radical.

$-\sqrt{\frac14}$

• A. $\frac12$
• B. $-\frac12$
• C. $-\frac{1}{16}$
• D. No real solution (Imaginary)

Evaluate the radical.

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

• A. $2.23$
• B. $5$
• C. $-5$
• D. No real solution (Imaginary)

Nth Roots

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 want to do here is just talk more generally about roots, and I'm going to show you that roots really just fall into 2 types of categories. And I'm going to show you the differences between these categories. Now let's get started. I'm going to actually get back to this information later on in the video. I'm just going to 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 backward 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 backward 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 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 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 number 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 going to 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 a number raised to the 7th power, and then I 7th root it, it's basically like I'm just cancelling it out 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.