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## Algebra (all content)

# 5th roots

5th roots.

## Want to join the conversation?

- how is 5th root connected to our lives?(49 votes)
- It isn’t really needed it and other insane roots(insane representing the power of these roots) came out of the question: how far can we go. Also powers of two are used in machine language of computers.(4 votes)

- When do students start to learn about 5th roots. If so, are they hard to understand, or are they easy to students when they recognize it as a big part of math?(11 votes)
- 5th roots aren't really hard to understand when you know how square roots and cube roots work, because they all follow the same basic principles. I don't know when students start to learn about this, but in my school, the teachers kind of expect you to know how to deal with any type of root (like the 8th root or 6th root) once you have learned about cube roots (again, because it all follows the same basic principles).(4 votes)

- But how do we solve 1/32 to the 5th root? It doesn't tell how to in the video but it came up in the problems.(3 votes)
- 32=16*2=4*4*1=2*2*2*2*2=2^5, so 1^5/2^5 = (1/2)^5, so fifth root is 1/2. If I can take the fifth root of 32, the fifth root of 1/32 should not be hard.(4 votes)

- in what point of life are we gonna need 5th roots(4 votes)
- Maybe not in life, but the most important is that we can learn it for a higher score on math quizzes.(1 vote)

- how is 5th root connected to our lives?(0 votes)
- Knowing how to deal with higher roots helps a lot when you enter calculus. So, treat everything you learn now as building blocks. Nothing is useless.(9 votes)

- how would you enter this in a calculator? i have a TI-83 plus if that makes any difference.(1 vote)
- Go to the MATH button, the first screen under MATH has the cubed root as 4:, and any root as 5:. So the fifth root of 32, put try 5 MATH 5: 32. This is on the 84, but the 83 should be the same.(6 votes)

- I pretty much know how it works but in the future, isn't using a calculator gonna be much more efficient?(3 votes)
- Yes, but you may see you can't do a 5th root problem on a normal calculator.(1 vote)

- Square root, cube root... hypercube root?(3 votes)
- this is a good video!(3 votes)
- is there any limits to the roots?(2 votes)
- Nope! But I don't think the bigger ones are useful at all. (there might be some obscure problem that they would be useful in, but nothing you need to worry about unless you want to be a mathematician or somehting)(3 votes)

## 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.