5th roots | Mathematics I | High School Math | Khan Academy

Let's see if we can calculate the fifth root of 32. So, like always, pause the video and see if you can figure this out on your own.

So, let's just remind ourselves what a fifth root is. So, if x is equal to the fifth root of 32, that's the same thing as saying that x to the 5th power is equal to 32. So, we have to find some number where if you take five of them and multiply them together, that you get 32.

There's a couple of ways to approach it, especially when you're dealing with these really higher order roots here. So, let me rewrite the fifth root of 32 here. One way is you could try to factor 32 and see if there are factors that show up five times.

32, you might immediately recognize as an even number, so it's going to be divisible by two. It's 2 * 16. 16 is 2 * 8. 8 is 2 * 4. 4 is 2 * 2. So, in this case, doing the factoring technique worked out well because we see that this is 2 * 2 * 2 * 2 * 2, or 2 to the 5th power. You could rewrite this as the fifth root of 2 to the 5th power, which is, of course, going to be equal to two.

2 to the 5th power is 32. Now, let's do another one; it's going to be a little bit harder. All right, let's say we want to take the fifth root of 243.

So now, a much, much larger number. There's a couple of ways to do this. One, you could try the factoring, although that's going to be harder now that it's a larger number, or you could do a little bit of trial and error. Doing higher roots without the aid of some type of calculator or something gets a little bit more complicated.

So here, if we wanted to do the factoring technique, we could say, all right, it's not divisible by two. I like to start with the smallest possible factor. So, it's not divisible by two; is it divisible by three? You might be familiar with the test to see if something is divisible by three. You add up the digits and see if that sum of the digits is divisible by three.

So, if I were to take 2 + 4 + 3, that is equal to 9, and so it is divisible by three. So, this is going to be equal to 3 * let's see, 3 goes into 240 80 times, and then 1, so 80, 1 times. And so, 81 is also divisible by three. I have a sense of where this is going now. It's 3 * 27, which is 3 * 9, which is 3 * 3.

So, using the factoring method, we're able to see that 3 to the 5th power is 243. So, the fifth root of 243 is equal to three.

Now, another way that you could have done it is a little bit of trial and error. We already know, well, we know that 1 to the 5th power is just going to be one. We know that 2 to the 5th power, we just calculated that—that's 32. Well, we now know what 3 to the 5th is.

But let's say we're just trying to zoom in on it a little bit. So, let's say if you wanted to see what four to the 5th is. Well, that would be four * 4 * 4 * 4 * 4. So, let's see, this is going to be 16. 16 * 4 is 64. 64 * 4 is 256, and then that times 4, and I just happen to know this, but you might want to do it by hand—this is 1024.

So, if you're taking the fifth root of 243, you're saying, hey, what to the fifth power—something to the fifth power is equal to 243. And if you have a sense that it's going to be an integer solution, if you think it's going to be something like a two or a three, well then, three is probably going to be a good guess here.

If the possible answers are going to be decimals, then it's going to be a lot more complicated. But that's another way to say, hey, maybe I'll try out three, and if you try it out, three, you would get 243.