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Simplifying square roots (variables)

A worked example of simplifying radical with a variable in it. In this example, we simplify 3√(500x³). Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user Steve Martella
    Hi all.
    Looking for some help with the following:

    3/√63

    So far I have:

    3/√63
    3/(√9 x √7)
    3/3√7
    √7

    The answer in the book I'm studying from, however says that the answer is:

    √7/7

    How does one arrive at this answer?
    (54 votes)
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    • piceratops ultimate style avatar for user Wall of Words
      Hi. When working with radicals, you can't have a radical for a denominator. This is not considered proper in Algebra. So your answer to
      3 / √63 would eventually equal 1/ √7.
      In order to make it algebraically proper, we have to multiply it by √7 / √7 (which is equal to one, so we know it won't change the value of our equation).
      So we get:
       1 / √7 * √7 / √7 =
      √7 / √7 * √7
      √7 / 7

      We know that √ 7 * √7 is the same thing as √ 7*7, or √49, which is 7. That gives us the final answer of √7 / 7 which can't be reduced any farther.
      Hope this helps.Sylvia.
      (108 votes)
  • blobby green style avatar for user Les Howles
    This is hard stuff and he seems to be blowing through it rapidly like it's just review. What is the previous video he keeps referring to in his commentary? The video I see before this one is entitled "Simplifying Square Roots" and he doesn't seem to cover a lot of this. Am I missing something?
    (53 votes)
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    • piceratops sapling style avatar for user Eliza
      I'm with you. Last vid is the same for me, "Simplifying Square Roots". I watched and understood all of the previous vids leading up to this one and he did blow through this one without slowing down to take the time to really explain why he's doing that. Yet he'll go over extremely simple concepts ad nauseam
      (52 votes)
  • spunky sam blue style avatar for user Matt Shull
    You confused me...Why can't X be a negative number?
    (37 votes)
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    • piceratops ultimate style avatar for user Woohoo
      coughs I'm late 8 years but whatever.
      Simple: you can't have negative rads. Use a calculator but a negative number like -1, and it will say "Error" or something like that.
      There are no two numbers that when you multiply them twice (like 2 • 2, aka 2^2), you cant get a negative number. Like a negative times, a negative is positive, and a positive times a positive is a positive.
      And you cant multiply two different numbers (negative and positive) in a rad because rads represent a number multiplied by itself to get that number inside the rad.
      (4 votes)
  • duskpin seedling style avatar for user Renishia Toney
    I don't understand most of this. I think i'm missing a video.
    (32 votes)
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  • leaf green style avatar for user Emily Manzerolle
    I'm sorry, but can we try one WITHOUT an x? I'm going into Grade 11 and never learned this. How do you simplify something like (sq root)45?

    In the practice, there are questions where c= sq. root 45 and they need a simplified radical. How do you do that?
    (19 votes)
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  • male robot donald style avatar for user 2017ColbyHermanowski
    what happens when the absolute value of a number turns out to be a negative, not a positive
    (4 votes)
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  • aqualine ultimate style avatar for user Shrey
    Why does Sal keep saying "the principal root of..." as opposed to "the square root of..."? At I heard him say "the square root of, or the principal root of..." so does that mean they're the same thing? Because it appears as if he sort of corrected himself.
    (19 votes)
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  • blobby green style avatar for user laura.adiletta
    If you watch these videos in the order the lessons are published for Algebra basics > foundations > square roots module, there are a lot of references in "Simplifying square roots (variables)" to the previous video ( and ). But the previous video is about fractions, not variables. I had to jump over to another module (Math > Algebra 1 > Exponents & radicals > Simplifying square roots) to find the video I THINK he's referring to, "Simplifying square root expressions".

    Can anyone at Khan Academy explain to me why this learning module would leave out lessons required for completing the content?
    (14 votes)
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  • blobby green style avatar for user Matthew Thompson
    I'm at a loss, The quiz is asking questions that the videos have no explanation of. I have x^9, but the video only explains how to factor it one time..
    (7 votes)
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    • aqualine ultimate style avatar for user AD Baker
      Matthew,

      This video is part of the Algebra I course and is not meant to stand alone. Many of the concepts applied here are explained in previous videos in this course and in the Pre-Algebra course.

      Take a look at the outline of Algebra I (https://www.khanacademy.org/math/algebra). You may want to go back to the previous videos in Rational exponents & radicals or to the Exponents, radicals, and scientific notation section of Pre-Algebra (https://www.khanacademy.org/math/pre-algebra)

      The Law of Exponents tells us

      x^a * x^b = x^(a+b)

      Therefore

      x^4 + x^5 = x^(4+5) = x^9

      x^4 is a perfect square - x^2 * x^2

      You could also consider

      x^9 = x * x * x * x * x * x * x * x * x

      x^9 = (x * x * x * x) * (x * x * x * x) * x

      x^9 = x^4 * x^4 * x
      (14 votes)
  • leaf blue style avatar for user Pi is the best
    Around of the video Sal mentions the absolute value of | x | which he got from the x^2 under the radical. if he were to use the absolute value for the x^2 why would he not use the absolute value of the lone | x | under the radical?
    (4 votes)
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    • piceratops ultimate style avatar for user Kanmani
      Because the lone x was not a perfect square, he could not simplify the radical. Only if you are taking the principle root and you are SIMPLIFYING the radical can you put in the absolute value. If he would have put the absolute value sign for the x under the radical it would've become:
      sqrt(|x|) = +/- (x^1/2)
      Which still gives you the negative root which is extraneous for principle roots. Therefore putting the absolute value under the radical is never done when taking the principle root as it yields unwanted answers.
      (6 votes)

Video transcript

What I want to do in this video is resimplify this expression, 3 times the principal root of 500 times x to the third, and take into consideration some of the comments that we got out on YouTube that actually give some interesting perspective on how you could simplify this. So just as a quick review of what we did in the last video, we said that this is the same thing as 3 times the principal root of 500. And I'm going to do it a little bit different than I did in the last video, just to make it interesting. This is 3 times the principal root of 500 times the principal root of x to the third. And 500-- we can rewrite it, because 500 is not a perfect square. We can rewrite 500 as 100 times 5. Or even better, we could rewrite that as 10 squared times 5. 10 squared is the same thing as 100. So we can rewrite this first part over here as 3 times the principal root of 10 squared times 5 times the principal root of x squared times x. That's the same thing as x to the third. Now, the one thing I'm going to do here-- actually, I won't talk about it just yet, of how we're going to do it differently than we did it in the last video. This radical right here can be rewritten as-- so this is going to be 3 times the square root, or the principal root, I should say, of 10 squared times the square root of 5. If we take the square root of the product of two things, it's the same thing as taking the square root of each of them and then taking the product. And so then this over here is going to be times the square root of, or the principal root of, x squared times the principal root of x. And the principal root of 10 squared is 10. And then what I said in the last video is that the principal root of x squared is going to be the absolute value of x, just in case x itself is a negative number. And so then if you simplify all of this, you get 3 times 10, which is 30-- and I'm just going to switch the order here-- times the absolute value of x. And then you have the square root of 5, or the principal root of 5, times the principal root of x. And this is just going to be equal to the principal root of 5x. Taking the square root of something and multiplying that times the square root of something else is the same thing as just taking the square root of 5x. So all of this simplified down to 30 times the absolute value of x times the principal root of 5x. And this is what we got in the last video. And the interesting thing here is, if we assume we're only dealing with real numbers, the domain of x right over here, the x's that will make this expression defined in the real numbers-- then x has to be greater than or equal to 0. So maybe I could write it this way. The domain here is that x is any real number greater than or equal to 0. And the reason why I say that is, if you put a negative number in here and you cube it, you're going to get another negative number. And then at least in the real numbers, you won't get an actual value. You'll get a square root of a negative number here. So if you make this-- if you assume this right here, we're dealing with the real numbers. We're not dealing with any complex numbers. When you open it up to complex numbers, then you can expand the domain more broadly. But if you're dealing with real numbers, you can say that x is going to be greater than or equal to 0. And then the absolute value of x is just going to be x, because it's not going to be a negative number. And so if we're assuming that the domain of x is-- or if this expression is going to be evaluatable, or it's going to have a positive number, then this can be written as 30x times the square root of 5x. If you had the situation where we were dealing with complex numbers-- and if you don't know what a complex number is, or an imaginary number, don't worry too much about it. But if you were dealing with those, then you would have to keep the absolute value of x there. Because then this would be defined for numbers that are less than 0.