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## Algebra (all content)

### Course: Algebra (all content) > Unit 11

Lesson 8: Radicals (miscellaneous videos)- Simplifying square-root expressions: no variables
- Simplifying square roots of fractions
- Simplifying rational exponent expressions: mixed exponents and radicals
- Simplifying square-root expressions: no variables (advanced)
- Intro to rationalizing the denominator
- Worked example: rationalizing the denominator
- Simplifying radical expressions (addition)
- Simplifying radical expressions (subtraction)
- Simplifying radical expressions: two variables
- Simplifying radical expressions: three variables
- Simplifying hairy expression with fractional exponents

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# Simplifying radical expressions (addition)

A worked example of simplifying an expression that is a sum of several radicals. In this example, we simplify √(2x²)+4√8+3√(2x²)+√8. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- at 2;40, why would x squared really be the absolute value of x?(73 votes)
- any number - either positive or negative - squared will always be positive -> therefore it will always be the absolute value of the number(8 votes)

- How did you know right off that they are asking for the principal square root? Is every square root aksing for the principal square root? What would be an alternative? What would it look like?(20 votes)
- The principle square root is simply the radical sign.

The NEGATIVE square root will have "-" before the radical sign.

If the problem wants both, it will have a "±" before the radical sign.(39 votes)

- I am not really sure where to put this, or find it ,but I was wondering if there are any lessons about dividing radicals?(11 votes)
- Dividing by a radical? While you could leave an expression in the form a/sqrt(b), it is more appropriate to multiply that by sqrt(b)/sqrt(b) to get (a*sqrt(b))/b. (This works since a number divided by itself is 1.(8 votes)

- At4:35wouldn't 4lxl*√2 + 10√ = 4lxl+11√2 (meaning wouldn't the 10√2 + √2 =11√2(8 votes)
- Not quite, because 4|x| is multiplied by √2, not added. If it was 4|x| + √2 + 10√2, you would be correct. If you are trying to factor out the √2, you have to add together the ENTIRE numbers being multiplied by √2. So you would add 10 and 4|x|. Since these are not like terms, you get:

(4|x| + 10) * √2(13 votes)

- is it possible that there can be a zero or a 1 behind the radical?(5 votes)
- Yes, but if 1 is under the square root sign, it just simplifies to 1 because the square root of 1 is 1. For zero, the square root of zero is zero so the whole term becomes zero.(4 votes)

- why does Sal say at4:30"principal square root"(4 votes)
- Because the √ means principal square root, not square root.(4 votes)

- At4:35the answer was: (4lxl+10)√2.

Why isn't the answer 14lxl√2?(5 votes) - if you have a denominator that is a whole number , and the numerator is a square root, what do you do?(4 votes)
- you have to take the square root symbol out of the denominator. You can do this by multiplying the numerator and the denominator by the denominator(3 votes)

- what happens if the two roots are different how do you simplify and then solve?(4 votes)
- (4|x|+10)sqrt2 ?

Can we factor out 2?

(2|x|+5)*2sqrt2 ?(4 votes)

## Video transcript

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