Hey everyone. Early in the course when we studied exponents, we saw how to square a number and we saw something like \(4^2\) 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 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 gonna be 1? Well, no. Because 1 multiplied by itself is 1. What about 2? Now, that just gets me 4. What about 3? \(3^2\), 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 two square roots, 3 and negative 3. This actually always works for positive real numbers. They always have two roots. There is a positive root like the 3, and textbooks sometimes call that the principal roots, but there's also a negative root, the negative 3. Alright? So, basically, if I start at 9 and I want to go backwards and take the square roots, there are two 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 there's a problem here. So if there are two possible answers for the square root of 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. 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 root, 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 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 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 radical 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. \(1^2\) is not going to be that because that's just 1. \(2^2\) is 4. \(3^2\) is 9. \(4^2\) 4 times 4 is 16. So I have to keep going. \(5^2\), which is 25. That's still not it. And what about \(6^2\)? Well, \(6^2\) is equal to 36. So it means all of these are wrong answers, but this one's the right one. I have 6. When I multiply by itself, it gets me 36. So which one is it but that also means that negative 36 oh, I'm sorry. Negative 6 also gets me to 36. So what's the answer here? Is it the positive or is it the negative? Remember, the radical is by itself, so this actually means it's just talking about 6, and it's not talking about both of them or the negative one. 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 root of 36. So this answer is negative 6 mplified by itself, gets you negative 36 or, you know, in this case, negative 9. Can I do that? Well, here, what happens is if I try to do 3, remember \(3^2\) is not negative 9. It's just positive 9. So that's not gonna work. And what about negative 3? That's also not gonna work because if I took negative 3 and squared it, I, you know, I just got a positive 9. So, in other words, that's not gonna work either because that just equals 9. 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. 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 memory tool to use. When we saw negatives that were outside of radicals, that was perfectly fine, and that was okay. So, for example, we saw negative outside of radical 9 or 36. That's perfectly fine. But if you see a negative inside, that means that it's imaginary. So outside is okay, but inside is imaginary. Alright? So negative radical 36 over here, perfectly fine, but the radical of negative 36, that's imaginary. Alright? 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.
Radical Expressions - Online Tutor, Practice Problems & Exam Prep
Created using AISquare roots and cube roots are fundamental concepts in mathematics, representing the inverse operations of squaring and cubing numbers. For positive numbers, square roots yield two solutions: a positive and a negative root, while cube roots provide only one solution, which retains the sign of the original number. Notably, negative numbers under even roots result in imaginary numbers, while odd roots can yield real negative results. Understanding these principles is crucial for solving equations involving radicals and exponents effectively.
Square Roots
Video transcript
Evaluate the radical.
No real solution (Imaginary)
Evaluate the radical.
No real solution (Imaginary)
Nth Roots
Video transcript
Welcome back, everyone. So we recently saw that squares and square roots are opposites of each other. What I'm going to show you in this video is that squaring isn't the only exponent we can do. We can also raise numbers to the 3rd, 4th power, and so on and so forth. What I want to do here is talk more generally about roots, and I'm going to show you that roots really fall into two 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 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 discussed square roots, we said that 22 was equal to 4, and −22 was also equal to 4. So both of these numbers were square roots of 4. That means that if you go backward from 4, if you undo that taking the square roots, you get 2 and you should get negative 2 as well. But what happens if I take 23? Well, let's just take a look here. 23 is 2 times 2, which is 4, and 4 times 2 is 8. Negative 2, when cubed, since the negatives cancel for the first two terms but then you have another factor of negative 2, this turns into negative 8. So here's the difference: When I took 2 and negative 2 and squared them, I got the same number, 4. Whereas, when I cube 2 and negative 2, I get different numbers, 8 and negative 8.
Just as the square root was the opposite of squaring, we can take cube roots as the opposite of cubing. And what we see here is that the cube root of 8 is not both numbers. You don't get two numbers because it only gets us back to 2, 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.
Moreover, what we also saw was 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 three times, gets you to negative 8.
More generally, if you take a number and 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. 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.
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 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, 7th roots, and stuff like that. So let's go ahead and take a look at some examples. Using these rules, we'll evaluate the following nth roots or indicate if the answer is imaginary. We will examine the 4th root of 81 and the 5th root of -32, and we will discover if an imaginary number appears with a negative inside and even index.
Thanks for watching. That's it for this one.
Here’s what students ask on this topic:
What is the square root of a negative number?
The square root of a negative number is considered imaginary. This is because no real number, when squared, results in a negative value. For example, the square root of -9 is written as √(-9), which is equal to 3i, where 'i' is the imaginary unit representing √(-1). In general, √(-a) = i√(a) for any positive number 'a'.
Created using AIHow do you simplify radical expressions?
To simplify radical expressions, follow these steps: 1) Factor the number inside the radical into its prime factors. 2) Pair the prime factors. 3) Move each pair of prime factors outside the radical as a single number. For example, to simplify √(72), factor 72 into 2 × 2 × 2 × 3 × 3. Pairing the factors, we get (2 × 2) and (3 × 3). Moving the pairs outside the radical, we get 2 × 3√(2) = 6√(2).
Created using AIWhat is the difference between square roots and cube roots?
Square roots and cube roots are both types of roots, but they differ in their properties. The square root of a number 'a' is a number 'b' such that b² = a. Square roots of positive numbers have two solutions: a positive and a negative root. Cube roots, on the other hand, are numbers 'b' such that b³ = a. Cube roots have only one solution, which retains the sign of the original number. For example, the square roots of 9 are 3 and -3, while the cube root of 8 is 2.
Created using AIHow do you find the nth root of a number?
To find the nth root of a number 'a', you are looking for a number 'b' such that bⁿ = a. The nth root is denoted as √[n](a). For example, the 4th root of 81 is a number 'b' such that b⁴ = 81. By factoring, we find that 3⁴ = 81, so the 4th root of 81 is 3. If the index 'n' is even and 'a' is negative, the result is imaginary. If 'n' is odd, the result retains the sign of 'a'.
Created using AIWhat are imaginary numbers and how are they used in radical expressions?
Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit 'i', where i = √(-1). They are used in radical expressions to represent the square roots of negative numbers. For example, √(-4) is written as 2i because 2i × 2i = -4. Imaginary numbers are essential in complex number theory and have applications in engineering, physics, and other fields.
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