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We're asked to add and simplify and we have the principle root of two x squared plus four times the principle

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root of eight plus three times the principle root two x squared plus the principle root of eight

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so we can do a little bit of adding, we can actually simplify

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first and then add or we can add first and then simplify

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but it looks like we can already add so lets try and do that

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so here, right over here, I have a principle root of two x squared and over here I have three principle

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roots of two x squared, well if I have one of something here and I have three of something here and i need

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to add them together I can put a one co-efficient out here to make it clear this is one of this thing

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and I have three of these things but if I have one of this thing and three more of these things and I add them together I am

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going to have four of those things, so this is four times the principle root of two x squared

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and that confuses a little bit, imagine that the whole principle root of two x squared was some variable

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lets say this whole thing was "a" and lets say that this whole thing was "a" as well, because its the same thing, you'd have one "a"

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plus three "a"'s which will give you four "a"'s, in this case "a" is all of this business right over here

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so we added those terms, and then we wanted to think about we have four principle roots of "a" and we

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have one more principle roots of "a", so same idea you have four of these things I am circling in magenta

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and you have one more of these things that I am circling in magenta, that one co-efficient is implicit

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so if I have four of something plus one more of something it becomes five of that something

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so plus plus five times the square root, plus five times the square root of eight

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and now we'll see if we can simplify this anymore, we have four of something and we have five of something

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else, so you can't just add these two things together, but maybe we can simplify this a little bit

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so we know that the principle root of two x squared, this is the same thing as, so let me write the four

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out front, so we have the four, and the principle root of two x squared is the same thing as the principle

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root of two times the principle root of x squared so I just rewrote this part over here

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and then we have plus five times, now eight can be written as a product of a perfect square and a

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not so perfect square, eight can be written as four times two, so lets write it that way

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so if we view this whole, this is the principle root, the square root of four times two, we can re-write this as the

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five times the square root of four, or the principle root of four times the principle root of two and

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what can we simplify here? well we know what the principle root of x squared is, it is the positive square

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root of x squared, so it is not just x, you might be tempted to say it is x but since we know it is

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the positive square root we have to say it is the absolute value of x, because what if x was negative?

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if was x was negative, you'd have , lets say it was negative three, you'd have negative three squared,

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you'd have a positive nine, and so the principle root of a positive nine is going to be a positive three

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and so it wouldn't just be x, it wouldnt be negative three, it would be positive three, so you have to

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take the absolute value, and the other thing that is a perfect square is the four right here, its principle

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root is two, its principle square root i should say is two, so now you have, if we just change the order we

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are multiplying right here, you have four, four times the absolute value

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of x, four times the absolute value of x, times the square root of two, times the square root of two,

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I want to do that in that same yellow color, times the square root of two, plus plus we have five times

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two, which is ten, right, this whole thing is simplified to two, so we have plus ten

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square roots of two, now we could call it a day, and say we are all done adding and simplifying

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or you could add a little bit more depending on how you wanna view it, because over here you have

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four times the absolute value of x square roots of two, and here you have ten square roots of two

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so you have four absolute value of x of something, and you have ten of that same something, you could

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add them up, or another way to think about it is, you could factor out a square root of two

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either one of those works, so you get four times the absolute value of x, plus ten plus ten times times

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the principle square root of two, so depending on whether you view this of this more simplified, one

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of those two will will satisfy you