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## Radicals

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# 5th roots

## Video transcript

- [Instructor] 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 x to the fifth power is equal to 32. So, we have to find some number where, if you take five of them
and multiply them together, you'd get 32. So, there is a couple of
ways to approach this. Especially when you're dealing with these really high order roots here. So, let me rewrite the
fifth root of 32 here. One way is you could try to factor 32 and see are there factors
that show up five times? So, 32 you might immediately
recognize is an even number. So, it's gonna be divisible by two. It's two times 16. 16 is two times eight. Eight is two times four. Four is two times two. So, in this case, doing the factoring technique worked out well. 'Cause we see that this is two times two times
two times two times two or two to the fifth power. You could rewrite this as the fifth root of two to the fifth power, which is, of course,
going to be equal to two. Two to the fifth power is 32. Now, let's do another one. It's gonna be a little bit harder. Let's say we wanna take
the fifth root of 243. So, now, a much, much larger number. So, there's a couple of ways to do this. One, you could try the factoring. Although, that's gonna 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 is a little bit more complicated. So, here, if we wanted to
do the factoring technique. We could say, alright,
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? And 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
two plus four plus three, that is equal to nine. And so it is divisible by three. So, this is going to be
equal to three times... Let's see three goes into
240 80 times and then one. So, 81 times. And so, 81 is also divisible by three. I have a sense of where this is going now. It's three times 27,
which is three times nine. Which is three times three. So, using the factoring method, we're able to see that three
to the fifth 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 one to the fifth power is just going to be one. We know that two to the fifth power... We just calculated that. That's 32. Well, we now know what
three to the fifth is. 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 fifth is. Well, that would be four times four times four times four times four. So, let's see, this is going to be 16. 16 times four is 64. Times four is 256. And then, that times four... And I just happen to know this. But you might wanna do it by hand. This is 1024. So, if you're taking the cube root of 243, you're saying 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 gonna be decimals, then it's going to be
a lot more complicated. But that's another way. Say, hey, maybe I'll try a three. And, if you try out
three, you would get 243.