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.