Simplifying radical expressions (subtraction)

we're asked to subtract all of this craziness over here and it looks daunting but if we really just focus it actually should be pretty straightforward to subtract and simplify this thing because right from the get-go I have 4 times the 4th root of 81 X to the 5th and from that I want to subtract 2 times the 4th root of 81 X to the 5th and so you really can just say look I have 4 of something and then there's something I'll just circle in yellow I have 4 of this it could be lemons I have 4 of these things and I want to subtract 2 of these things these are the exact same things they're the 4th root of 81 X to the 5th 4th root of 81 X to the 5th so if I have 4 of if I have 4 lemons and I want to subtract 2 lemons I'm going to have 2 lemons left over or if I have 4 of this thing and I take away 2 of this thing I'm going to have 2 of these things left over so these terms right over here simplify to 2 times the fourth root of 81 X to the 5th and I got this two just by subtracting the coefficients 4 of something minus 2 of something is equal to 2 of that something and then that of course we still have this minus the regular principle square root of x to the 3rd of X to the third now I want to try to simplify I want to try to simplify what's inside of these under the radical signs so that we can on this in this example actually take the fourth root and over here actually take maybe a principal square root so first of all let's see if 81 either is a is something to the 4th power at least can be factored into something that is a something to the 4th power so 81 if you do prime factorization is 3 times 27 27 is 3 times 9 and 9 is 3 times 3 so 81 is exactly 3 times 3 times 3 times 3 so 81 actually is 3 to the fourth power which is convenient because we're going to be taking the 4th root of that and then X to the fifth we can write as a product we can let me write it over here so it doesn't get messy so I'm going to write what's under the radical as 3 to the 4th power x times X to the fourth power times X X to the fourth times X is X to the fifth power and I'm taking the fourth root of all of this and taking the fourth root of all of this that's the same thing as taking the fourth root of this as taking the fourth root of this now let me just I'm going to want to skip steps so I'm taking the fourth root I'm taking the fourth root of all of it right over there and of course I have a two out front and then X to the third can be written as x squared times X so it's minus the principal square root of x squared times X and I broke it up like this because this right over here is a perfect square now how can we simplify this a little bit and you're probably getting used to the pattern this is the same thing as the fourth root of 3 to the fourth times the fourth root of x to the fourth times the fourth root of x so let's just skip straight to that so what is what is the fourth root well I can write it let me write it explicitly although you wouldn't have to necessarily do this this is the same thing as the fourth as the fourth root of 3 to the fourth times the fourth root of x to the fourth times the fourth root of x times the fourth root of x and 2 is being multiplied times all of that and then this over here is minus the principal square root of x squared times the principal square root of x and so if we try to simplify it the fourth root of 3 to the fourth power is just three so we get a three there the fourth root of x to the fourth power is just going to be X is just going to be is just actually but I just reminded myself we have to be careful there it is not just X because what if X is negative if X is negative then X to the fourth power is going to be a positive value and when you take the fourth remember this is the fourth principal root you're going to get the positive version of X or really going to get the absolute value of X so here you're going to be getting you're going to be getting the absolute value of x and then although well you could make an argument that X needs to be positive if this thing is going to be well defined in the real numbers because then what's under the radical has to be positive but let's just go with this for right now and then we have the fourth of X and then over here the principle square root of x squared by the same logic by the same logic is going to be the absolute value of X and then this is just the principal square root of x so let's multiply everything out we have 2 times 3 times the absolute value of X so 2 times 3 is 6 times the absolute value of x times the principal or the the principal fourth root of x I should say minus minus we took out the absolute value of x times the principal root of x and we can't do any more subtracting just because you have to realize this is a fourth root this is a regular square root principle square root if these were the same root then maybe we could simplify this a little bit more and so then we are all done and we have fully simplified it and if you make the assumption that this is defined for real numbers so that the domain over here this what has to be under these radicals has to be positive actually in every one of these cases and if there need to be positive if we're not going to be dealing with imaginary numbers all of these need to be positive their domains our X has to be greater than or equal to zero then you could assume that the absolute value of x is the same as X but I'll just stick it right here if you restrict the domain you could get rid of the absolute value signs