Simplifying higher-index roots

so far when we were dealing with radicals we've only been using the square root we've seen that if I write a radical sign like this and put a nine under it this means the principal square root of nine which is positive three or you could view it as the positive square root of nine now what's implicit when we write it like this is that I'm taking the square root so I could have also written it like this I could have also written the radical sign like this and written this index two here which means the square root the principal square root of nine find me something that if I square that something I get nine and the radical sign doesn't just have to apply to a square root it can you can change the index here and then take an arbitrary root of a number so for example if I were to ask you what you could imagine this is called the cube root or you could call it the third root of 27 what is this well this is some number that if I take it to the third power I get 27 well the only number that if you take it to the third power you get 27 is equal to 3 right 3 times 3 times 3 3 times 3 times 3 is equal to 27 9 times 3 27 so let's do a couple of so likewise let me just do one more just just so if I have 16 doing a different color if I have 16 and I want to take the fourth root of 16 what number times itself 4 times is equal to 16 and if it doesn't pop out at you immediately you can actually just do a prime factorization of 16 to figure it out see 16 is 2 times 8 8 is 2 times 4 4 is 2 times 2 so this is equal to the fourth root of 2 times 2 times 2 times 2 right you have these four 2's here well I have 4 2 is being multiplied so the fourth root of this must be equal to must be equal to 2 and you could also view this as kind of the fourth principal root because if these were all negative twos it would also work or so it you know there there's multiple ways that you there's just like you have multiple square roots you have multiple fourth roots but the radical sign implies the principal root now with that said we've simplified traditional square roots before now we should hopefully be able to simplify radicals with higher power roots so let's try a couple let's say I want to simplify this expression the fifth root of 96 so like I said before let's just factor this right here so 96 is two times 48 which is two times 24 which is two times 12 which is 2 times 6 which is 2 times 3 so this is equal to the fifth root of 2 times 2 times 2 times 2 times 2 2 times 2 times 2 times 2 times 2 times 3 times 3 or another way you could view that is you could view it to a fractional power you could view it to a fractional power we've talked about that already this is the same thing as 2 times 2 times 2 times 2 times 2 times 3 to the 1/5 power taking let me make this clear taking having an nth root of some number is equivalent to taking that number to the 1 over N power these are equivalent statements right here so if you're taking this to the 1/5 power this is the same thing as taking 2 times 2 times 2 times 2 times 2 to the 1/5 times 3 to the 1/5 now I have something that's multiplied I have 2 multiplied by itself 5 times and I'm taking that to the 1/5 well the 1/5 power of this is going to be 2 or the 5th root of this is just going to be 2 so this is going to be 2 right here this is going to be 3 to the 1/5 power 2 times 3 to the 1/5 which which is is simplified about as much as you can simplify it but if we want to keep well in radical form we could write it as two times the fifth root the fifth root of three just like that let's try another one let's try another one let's say we wanted to let me put some variables in there let's say we wanted to simplify the sixth root of 64 times X to the eighth so let's do 64 first 64 is equal to 2 times 32 which is 2 times 16 which is 2 times 8 which is 2 times 4 which is 2 times 2 so we have one two three four five six so it's essentially 2 to the sixth power so this is equivalent to the sixth root of 2 to the sixth that's what 64 is times X to the 8th power now the sixth root of 2 to the sixth that's pretty straightforward that's going to be so this is going to be equal to this part right here is just going to be equal to 2 it's going to be 2 times the sixth root the sixth root the sixth root of x to the eighth power X to the 8th power and how can we simplify this well X to the 8th power that's the same thing as X to the sixth power times x squared right you have the same base you would add the exponents this is the same thing as X to the 8th so this is going to be equal to 2 times the sixth root of x to the sixth times x squared and the sixth root this part right here the sixth root of x to the sixth that's just X so this is going to be equal to 2 times X times the sixth root of x squared now we can simplify this even more if you really think about it remember this expression right here this is the exact same thing as x squared to the 1/6 power and if you remember your exponent properties this is this when you when you raise something to exponent and then raise that to an exponent that's equivalent to X to the 2 times 1/6 power or let me write this 2 times 1/6 power which is the same thing let me not forget to write my 2x there so I have a 2x there and a 2x there and this is the same thing as 2x I just the same 2x there times X to the X to the 2 6 or if we want to write that in simple simple form or lowest common form you get 2x times X to the what do you have here X to the 1/3 so if you want to write it in radical form you could write this as equal to 2 times 2x times the third root of x or the other way to think about it you could just say so we could just go from this point right here so we have we could write this we could ignore this what we did before we could say this is the same thing as 2 times X to the 8th to the 1/6 power right X to the 8th to the 1/6 power so this is equal to 2 to the 2 times X to the 8 times 1 6 8 6 power now we can we can reduce that fraction that's going to be 2 times X to the 4/3 power and this and this are completely equivalent why is that because we have 2 times X or 2 times X to the first power times X to the 1/3 power you add 1/2 1/3 you get 4/3 so hopefully you found this little tutorial on higher power radicals interesting I think it is useful to kind of see it in prime factored form and realize oh if I'm doing the 6 root I have to find a prime factor that shows up at least 6 times and I can figure out that's 2 to the 6th anyway hopefully you found this mildly useful