5th roots

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 fifth 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 so there is a couple of ways to approach it especially when you're doing 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 is are there factors that show 5 times and so 32 you might immediately recognize as an even number so it's going to be divisible by 2 it's 2 times 16 16 is 2 times 8 8 is 2 times 4 4 is 2 times 2 so in this case doing the factoring technique worked out well because we see that this is 2 times 2 times 2 times 2 times 2 or 2 to the fifth power you could rewrite this as the fifth root of 2 to the fifth power which is of course going to be equal to 2 2 to the 2 to the fifth 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 so there's a couple of ways to do this one you could try the factoring although it's going to be harder now that it's a larger number or you could do a little bit of trial and error you doing higher roots without the aid of some type of calculator or something it's 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 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 3 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 3 times 27 which is 3 times 9 which is 3 times 3 so using the factoring method we're able to see that 3 to the fifth power is 243 so the fifth root of 243 is equal to 3 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 fifth power is just going to be 1 we know that 2 to the fifth power we just calculated that that's 32 well we now know what 3 to the fifth 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 4 to the fifth is well that would be 4 times 4 times 4 times 4 times 4 so let's see this is going to be 16 16 times 4 is 64 times 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 10,000 and 24 so if you're taking the cube 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 a integer solution if you think it's going to be something like a 2 or 3 well then 3 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 say maybe I'll try all three and if you try it out 3 you would get 243