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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.

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  • starky ultimate style avatar for user Soloman
    okay so how does 180 times 1/2 = the sqrt of 90 pls help
    (12 votes)
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  • starky seedling style avatar for user Ivanov
    O.k, I really have no idea how this works; would someone please help me?
    (7 votes)
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    • leafers ultimate style avatar for user Jason Berg
      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)
  • blobby green style avatar for user gna0905
    By , 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)
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    • stelly blue style avatar for user Kim Seidel
      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)
  • leaf blue style avatar for user Buck
    I don't understand the fraction part.
    (5 votes)
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  • piceratops ultimate style avatar for user SarveshR
    I need help.
    (4 votes)
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  • piceratops tree style avatar for user Nox
    Why are we learning to simplify square roots that require factoring in algebra basics while factoring isn't covered until Algebra 1?
    (7 votes)
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  • blobby green style avatar for user emwodh tariku
    what if its 4.32 instead of 2.64 in the last question
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      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)
  • blobby green style avatar for user Lidia
    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)
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    • stelly blue style avatar for user Kim Seidel
      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)
  • aqualine ultimate style avatar for user 12abhik14qartz
    Why can't you just say √-40 instead of -√40?
    (3 votes)
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  • blobby green style avatar for user Kelsey Spencer
    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)
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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.