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we're asked to rationalize and simplify this expression right over here and like many problems there's multiple ways to do this we could simplify a little bit then rationalize and then simplify a little bit more or we could just rationalize and simplify and just to make sure that you know what they're even talking about rationalize is just a fancy word fancy way of saying we don't want to see any square roots of numbers in the denominator that's all it says so try to get these things outside of the denominator so the first thing that we can do let me simplify a little bit and then rationalize and then we could think about other ways to do it so what I'd like to do first is say well square the principal square root of 8 that can be simplified a little bit because 8 is the same thing so that's is the same thing as the square root of 4 times 2 which is the same thing as the square root of 4 times the square root of 2 so we can rewrite this entire expression as the numerator is still the same 16 plus 2x squared all of that over we can rewrite this as the square root of 4 over times the square root of 2 and the square root of the principal square root of 4 we know is just 2 so square root of 8 we can rewrite as 2 times the principal square root of 2 2 times the principal square root of 2 and I've simplified a little bit I've done no rationalizing just yet it looks like there's a little bit more simplification I can do first because everything in the numerator and everything in the denominator is divisible by 2 so let's divide the numerator by 2 so if you divide the numerator by 2 16 divided by 2 or you could view it as we're multiplying the numerator and the denominator by 1/2 so 16 16 times 1/2 is 8 2 x squared times 1/2 is just x squared and then 2 times the principal square root of 2 times 1/2 is just 1 or it's just sorry it's just a square root of 2 it's just 1 square roots of 2 so this whole thing has simplified to 8 plus x squared all of that over the square root of 2 and now let's rationalize this and the best way to get this radical out of the denominator is just multiply the numerator and the denominator by the principle square root of 2 so let's do that so times the principal square root of 2 over the principle square root of two now just to show that it works on the denominator what is the principle square root of two times the principal square root of two well it's going to be two and in our numerator we have eight we're going to distribute this onto both terms in this expression so you have eight times the principal square root of two plus plus the square root of two x times x squared and we can consider this done we have simplified the expression or if you want you could break it up you could say that this is the same thing as eight square roots of 2 over 2 which is 4 square roots of 2 plus square root of 2x squared over 2 so plus square root of 2 over 2x squared so depending on your tastes you might view this is more simple or this is more simple but though both are equally valid now I said there's multiple ways to do this we could have rationalized right from the get-go let me start with our original problem so our original problem was 16 plus 2x squared all of that over the principle square root of 8 we could have rationalized from the get-go by multiplying the numerator and the denominator by the principal square root of 8 multiplied by the principal square root of 8 and so in our denominator we'll just get 8 and then in our numerator we would get 16 times the principal square root of 8 plus 2 times the principal square root of 8x squared and now we can try to simplify this a little bit more you could say well everything is divisible by everything in the numerator and the denominator is divisible by 2 so the 16 could become an 8 if you divide by 2 the 2 becomes a 1 and then this 8 becomes a 4 and then you get 8 square roots of 8 plus square roots are e 8 square roots of 8 8 square roots of 8 plus the square root of 8 square root of R 8 8 x squared and then all of this all of this over 4 and you say wait this still looks kind of different than what we had here and the reason is we still haven't simplified this radical we know that we can rewrite the principal square root of 8 as 2 square roots of 2 this is two square roots of 2 and then we can see again that everything in the numerator and the denominator is also divisible by two so let's do that again so if you divide everything in the numerator by 2 you can get rid of this 2 that 2 and everything in the denominator by 2 this will become a 2 so then you have 8 square roots of 2 you have 8 square roots of 2 Plus this guy this is just a 1 now plus square root of 2x squared all of that over 2 which is exactly what we had gotten here