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### Course: Algebra basics > Unit 1

Lesson 4: Square roots- Intro to square roots
- Square roots of perfect squares
- Square roots
- Simplifying square roots
- Simplifying square roots of fractions
- Simplify square roots
- Simplifying square-root expressions: no variables
- Simplifying square roots (variables)
- Simplify square roots (variables)

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# Simplifying square-root expressions: no variables

When square roots have the same value inside the radical, we can combine like terms. First we simplify the radical expressions by removing all factors that are perfect squares from inside the radicals. Then we can see whether we can combine terms or not. If there is only one term, there is nothing to combine.

## Want to join the conversation?

- okay so how does 180 times 1/2 = the sqrt of 90 pls help(12 votes)
- Because 180*.5= 90, so finding the square root, it would be the sqrt of 90.(5 votes)

- O.k, I really have no idea how this works; would someone please help me?(7 votes)
- All you're really looking for are square numbers that can be pulled out of the radical.

An important thing to realize is that sqrt(a•b) = sqrt(a)•sqrt(b). This allows us to separate the radical expression into it's factors. If it has any square factors, they simplify, and you're left with a simplified expression.

Here's an example with actual numbers:

sqrt(12) = sqrt(4•3) = sqrt(4)•sqrt(3) = 2sqrt(3)(20 votes)

- By6:11, you can see the final answer, but I got 6√6 over 9. I got this by simplifying √128, then multiplying the whole fraction by √27 because a radical sign should never be on the denominator. Then after some simplifying, I got 6√6 all over the denominator 9. But I don't understand what I got wrong. Please help!(3 votes)
- I'm going to try and repeat your steps...

1)`√128 = 8 √2`

(I think your error could be here)

2)`8 √2 / √27 * (√27/√27)`

=`8 √54 / 27`

3)`8 √54 / 27`

=`8 √(9*6) / 27`

=`8*3 √6 / 27`

=`8 √6 / 9`

Either you didn't get the "8" out of √128, or you lost it somewhere along the way.

A couple of tips:

1) Try to reduce the fraction 1st. You can usually save your self quite a bit of work.

2) You did a better job then Sal in trying to get to a complete answer. Sal's answer would typically be considered incomplete as he didn't rationalized the denominator.

Any way, hope this helps.(9 votes)

- I don't understand the fraction part.(5 votes)
- Why are we learning to simplify square roots that require factoring in algebra basics while factoring isn't covered until Algebra 1?(7 votes)
- Please post questions once and be patient. Questions are answered by other KA users in their spare time. Prime factorization is a pre-algebra topic: https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples

Maybe you need to backup and review basic math and pre-algebra before doing Algebra.(0 votes)

- what if its 4.32 instead of 2.64 in the last question(2 votes)
- The last problem has 2 times 64, not a decimal number of 2.64. And, I'm assuming you means 4*32, not the decimal number 4.32. Use the * (shift 8) for a raised dot to indicate multiplication when typing.

Sal started with 128. He split it into 2(64) because 64 is the largest perfect square. And, all perfect square factors must be simplified out of the square root. If you start with 4(32), you can take the square root of 4, but you can leave the problem as 2 sqrt(32). 32 still contains factors that are perfect squares. You need to find all of them. You could do 2(16) and do the square root of the 16 as well as your original 4. If you don't see the 16, take out another 4, you'll have 128 = 4*4*8. Then split up the 8 to get 128 = 4*4*4*2. Each of those 4's is a perfect square and needs to be simplified by taking their square roots.

Hope this helps.(4 votes)

- 40 is also divisible by 2,but you also said 4 is divisible by 40,dose it matter,is the answer going to be same?(2 votes)
- 40 does not divide evenly into 4. I think you meant to say "40 is divisible by 4".

When simplifying square roots, you need to find perfect square factors and take their square root. You leave any factors inside the radical that are not perfect squares.

Yes, you can start by dividing 40 by 2, but 2 is not a perfect square. So, you need to keep factoring. If needed, you can factor down to prime factors: 40 = 2*2*2*5. Any pair of matching factors creates a perfect square. Over time, you would recognize the 4 is the perfect square factor, and you would factor out the 4 diretly.

Using the prime factors: 2*2*2*5, split them up. (2*2)(2*5). The (2*2) is the perfect square. So, you can take its square root: sqrt(2*2) = 2. And, you would leave (2*5) inside the radical. This makes the final answer:

sqrt(40) = sqrt(2*2) sqrt(2*5) = 2 sqrt(10)

Hope this helps.(4 votes)

- Why can't you just say √-40 instead of -√40?(3 votes)
- Um... I thought this was supposed to be a video about solving square roots with variables... why is it called that when that’s not what the topic is about?(3 votes)

## Video transcript

- [Voiceover] Let's get
some practice rewriting and simplifying radical expressions. So in this first exercise,
and these are all from Khan Academy. It says simplify the expression
by removing all factors that are perfect squares
from inside the radicals, and combine the terms. If the expression cannot be
simplified, enter it as given. All right, let's see what we can do here. So, we have negative 40 (laughs), the negative square
root of 40 I should say. Let me write a little bit
bigger so you can see that. So the negative square root of 40 plus the square root of 90. So let's see, what
perfect squares are in 40? So, what immediately jumps out at me is that this, it's divisible by four and four is a perfect square. So this is the negative square root of four times 10, plus the square root of, well
what jumps out at me is that this is divisible by nine. Nine is a perfect
square, so nine times 10. And if we look at the 10s
here, 10 does not have any perfect squares in it anymore. If you wanted to do a
full factorization of 10, a full prime factorization, it would be two times five. So there's no perfect squares in 10. And so we can work it out from here. This is the same thing
as the negative of the square root of four times
the square root of 10, plus the square root of nine, times the square root of 10. And when I say square
root, I'm really saying principal root, the positive square root. So it's the negative of the
positive square root of four, so that is, so let me do
this is in another color. so it can be clear. So, this right here is two. This right here is three. So it's going to be equal to negative two square roots of 10 plus three square roots of 10. So if I have negative two of something and I add three of that
same something to it, that's going to be what? Well that's going to be
one square root of 10. Now this last step
doesn't make full sense. Actually, let me slow
it down a little bit. I could rewrite it this way. I could write it as
three square roots of 10 minus two square roots of 10. That might jump out at
you a little bit clear. If I have three of something
and I were to take away two of that something, and that case it's squares of 10s, well, I'm
going to be left with just one of that something. I'm just gonna be left
with one square root of 10. Which we could just write
as the square root of 10. Another way to think about it is, we could factor out a
square root of 10 here. So you undistribute it, do the distributive property in reverse. That would be the square root of 10 times three minus two, which is of course, this is just one. So you're just left with
the square root of 10. So all of this simplifies
to square root of 10. Let's do a few more of these. So this says, simplify
the expression by removing all factors that are
perfect squares from inside the radicals, and combine the terms. So essentially the same idea. All right, let's see what we can do. So, this is interesting. We have a square root of 1/2. So can I, well actually, what could be interesting
is since if I have a square root of something
times the square root of something else. So the square root of 180 times the square root of 1/2, this is the same thing as the square root of 180 times 1/2. And this just comes straight
out of our exponent properties. It might look a little bit
more familiar if I wrote it as 180 to the 1/2 power,
times 1/2 to the 1/2 power, is going to be equal to 180 times 1/2 to the 1/2 power, taking the square root, the principal root is the same
thing as raising something to the 1/2 power. And so this is the square
root of 80 times 1/2 which is going to be
the square root of 90, which is equal to the square root of nine times 10, and we just
simplified square root of 90 in the last problem, that's
equal to the square root of nine times the square root
of, principle root of 10, which is equal to three
times the square root of 10. Three times the square root of 10. All right, let's keep going. So I have one more of these examples, and like always, pause the video and see if you can work
through these on your own before I work it out with you. Simplify the expression
by removing all factors that are perfect square,
okay, these are just same directions that we've
seen the last few times. And so let's see. If I wanted to do, if I
wanted to simplify this, this is equal to the square root of, well, 64 times two is 128, and 64 is a perfect square,
so I'm gonna write it as 64 times two, over 27 is nine times three. Nine is a perfect square. So this is going to be the same thing. And there's a couple of ways
that we can think about it. We could say this is the same thing as the square root of 64 times two, over the square root of nine times three, which is the same thing
as the square root of 64 times the square root of two, over square root of nine
times the square root of three, which is
equal to, this is eight, this is three, so it would be
eight times the square root of two, over three times
the square root of three. That's one way to say it. Or we could even view the
square root of two over the square root of three
as a square root of 2/3. So we could say this is eight over three times the square root of 2/3. So these are all possible ways of trying to tackle this. So we could just write it, let's see. Have we removed all factors
that are perfect squares? Yes, from inside the radicals and we've combined terms. We weren't doing any
adding or subtracting here, so it's really just
removing the perfect squares from inside the radicals
and I think we've done that. So we could say this is
going to be 8/3 times the square root of 2/3. And there's other ways
that you could express this that would be equivalent but hopefully this makes some sense.