Adding and simplifying radicals | Exponent expressions and equations | Algebra I | Khan Academy

Adding and Simplifying Radicals

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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

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

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

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

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

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

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

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

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

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

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

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"

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

of those two will will satisfy you