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Intro to rationalizing the denominator

When we have a fraction with a root in the denominator, like 1/√2, it's often desirable to manipulate it so the denominator doesn't have roots. To do that, we can multiply both the numerator and the denominator by the same root, that will get rid of the root in the denominator. For example, we can multiply 1/√2 by √2/√2 to get √2/2. Created by Sal Khan and CK-12 Foundation.

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  • blobby green style avatar for user Joshua Flynn
    Is it possible to rationalize a/pi?
    (54 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      So the question is really, "why can we use this method with some irrational numbers like 1/√2 and 1/√111 but not with other irrational numbers like 1/pi?"

      The difference is that you can use this technique for numbers under a radical that can be multiplied by other numbers under a radical to produce a whole number. √2 is irrational because there is no nice clean number multiplied by itself to become 2--the square root of 2 goes on forever like pi.
      √2 = 1.41421356237...........
      Same with many other numbers, both primes and composites. √5 and √38 are two more examples, as is the cube root of 25, ³√25. But if you multiply the square root of 2 times the square root of 2, you just get 2. And THAT is when we can use this method.

      So there is a limit to the cases where you can use it. If you try to multiply pi by itself, this magic doesn't happen--you still have a mess of decimal places and an irrational result in the denominator.
      (52 votes)
  • blobby green style avatar for user Lauren Miller
    - difference of squares?
    (5 votes)
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  • blobby green style avatar for user arsee99
    he gets this answer -24-12√5 for the third question. how are there 2 negative signs? shouldn't there be only 1 negative sign from the -1?
    (8 votes)
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  • leaf yellow style avatar for user Zach
    What happens if you have a denominator that happens to be a transcendental number? You can't rationalize it, so is there anything you can do with it?
    (7 votes)
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    • leaf blue style avatar for user Stefen
      It is perfectly fine to leave transcendental numbers in the denominator.

      Actually, rationalizing the denominator was important in the days when there were no calculators and you had to look up the values of irrational numbers in a table and then do the math. Since multiplication by hand is easier than division by hand, especially when dealing with irrationals, which need lots of digits to maintain accuracy, the practice of rationalizing the denominator (to put the irrational number in the numerator) was developed. Even though we have calculators now, it is considered good form to always rationalize denominators when you can.
      (5 votes)
  • hopper cool style avatar for user Thomasa
    How to rationalize cube root denominators?
    (5 votes)
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    • duskpin ultimate style avatar for user Jarrett Haroldsen
      Great question! Cube Root/nth root denominators can be rationalized using a very similar method to square root denominators. All you need to do is multiply both the top and bottom of the fraction by the Cube Root/nth root of the radicand (stuff inside of the radical) to the power of the index (3 for cube root denominators).

      For example, we can simplify 1/cubeRoot(2), by multiplying both the top and the bottom by cubeRoot(2^2) which is equal to cubeRoot(4)/cubeRoot(8)=cubeRoot(4)/2.

      Hope this helped!
      (4 votes)
  • blobby green style avatar for user Malachi Jones
    How do you solve rationalize a denominator when there are two radicals? An example would be 1 / (1 + √3 - √5)
    (4 votes)
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    • old spice man green style avatar for user jmascaro
      Good question Malachi,
      There may be an easier way but the way I figured it out takes two steps because of the three term denominator.
      1) Switch the plus and minus signs of the denominator then multiply giving you
      1 / (1 + √3 - √5) * ((1 - √3 + √5) / (1 - √3 + √5))
      2) After distribution, the denominator simplifies to
      -7 + 2√3√5
      so the fraction we have so far is
      (1 - √3 + √5) / (-7 + 2√3√5)
      3) We still have radicals in the denominator so we repeat step 1
      ((1 - √3 + √5) / (-7 + 2√3√5)) * ((-7 - 2√3√5) / (-7 - 2√3√5))
      4) The denominator simplifies to -11. The numerator is kind of gnarly but the entire fraction now looks like:
      (-7 - 3√3 - √5 - 2√3√5) / -11
      You may want to double check my math but I'm pretty sure I copied my paper and pencil notes correctly.
      Hope this helps!
      (6 votes)
  • hopper cool style avatar for user Atar137h
    Do you have to rationalize the numerator if it is irrational like √2?
    (2 votes)
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  • piceratops ultimate style avatar for user Suryaansh Chawla
    What if we have 3 or more denominators which are irrational?
    (3 votes)
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  • blobby green style avatar for user kturner416
    i need help if it has an exponent and a variable not clean cut numbers
    (2 votes)
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    • mr pants teal style avatar for user Tim
      I'm not 100% sure what you mean, but I'll do my best.
      Case 1: Variable in the numerator: x²/√7
      As you may guess, multiply by √7/√7 gives (√7x²)/7
      Not much of an improvement, but a rational denominator.

      Case 2: Variable in the denominator: 7/√x
      Multiply by √x/√x
      Now we have (7√x)/x
      Assuming that x is a rational number (not pi or something like that), we are done.
      Can we get x out of the denominator all together? Not really. It is debatable how much of an improvement this is.

      Case 3: As 2, but nasty! 7/(√x+2)
      Multiply by (√x-2).
      By switching the sign, we get in the denominator:
      (√x+2)(√x-2) = √x*√x + 2√x -2√x -4 = x - 4
      The numerator is 7(√x-2) = 7√x - 14
      Result: (7√x - 14) / (x - 4)
      (2 votes)
  • blobby green style avatar for user michaela.moye1130
    So is you have 12/6√13 how would that work
    (2 votes)
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Video transcript

In this video, we're going to learn how to rationalize the denominator. What we mean by that is, let's say we have a fraction that has a non-rational denominator, the simplest one I can think of is 1 over the square root of 2. So to rationalize this denominator, we're going to just re-represent this number in some way that does not have an irrational number in the denominator. Now the first question you might ask is, Sal, why do we care? Why must we rationalize denominators? And you don't have to rationalize them. But I think the reason why this is in many algebra classes and why many teachers want you to, is it gets the numbers into a common format. And I also think, I've been told that back in the day before we had calculators that some forms of computation, people found it easier to have a rational number in the denominator. I don't know if that's true or not. And then, the other reason is just for aesthetics. Some people say, I don't like saying what 1 square root of 2 is. I don't even know. You know, I want to how big the pie is. I want a denominator to be a rational number. So with that said, let's learn how to rationalize it. So the simple way, if you just have a simple irrational number in the denominator just like that, you can just multiply the numerator and the denominator by that irrational number over that irrational number. Now this is clearly just 1. Anything over anything or anything over that same thing is going to be 1. So we're not fundamentally changing the number. We're just changing how we represent it. So what's this going to be equal to? The numerator is going to be 1 times the square root of 2, which is the square root of 2. The denominator is going to be the square root of 2 times the square root of 2. Well the square root of 2 times the square root of 2 is 2. That is 2. By definition, this squared must be equal to 2. And we are squaring it. We're multiplying it by itself. So that is equal to 2. We have rationalized the denominator. We haven't gotten rid of the radical sign, but we've brought it to the numerator. And now in the denominator we have a rational number. And you could say, hey, now I have square root of 2 halves. It's easier to say even, so maybe that's another justification for rationalizing this denominator. Let's do a couple more examples. Let's say I had 7 over the square root of 15. So the first thing I'd want to do is just simplify this radical right here. Let's see. Square root of 15. 15 is 3 times 5. Neither of those are perfect squares. So actually, this is about as simple as I'm going to get. So just like we did here, let's multiply this times the square root of 15 over the square root of 15. And so this is going to be equal to 7 times the square root of 15. Just multiply the numerators. Over square root of 15 times the square root of 15. That's 15. So once again, we have rationalized the denominator. This is now a rational number. We essentially got the radical up on the top or we got the irrational number up on the numerator. We haven't changed the number, we just changed how we are representing it. Now, let's take it up one more level. What happens if we have something like 12 over 2 minus the square root of 5? So in this situation, we actually have a binomial in the denominator. And this binomial contains an irrational number. I can't do the trick here. If I multiplied this by square root of 5 over square root of 5, I'm still going to have an irrational denominator. Let me just show you. Just to show you it won't work. If I multiplied this square root of 5 over square root of 5, the numerator is going to be 12 times the square root of 5. The denominator, we have to distribute this. It's going to be 2 times the square root of 5 minus the square root of 5 times the square root of 5, which is 5. So you see, in this situation, it didn't help us. Because the square root of 5, although this part became rational,it became a 5, this part became irrational. 2 times the square root of 5. So this is not what you want to do where you have a binomial that contains an irrational number in the denominator. What you do here is use our skills when it comes to difference of squares. So let's just take a little side here. We learned a long time ago-- well, not that long ago. If you had 2 minus the square root of 5 and you multiply that by 2 plus the square root of 5, what will this get you? Now you might remember. And if you don't recognize this immediately, this is the exact same pattern as a minus b times a plus b. Which we've seen several videos ago is a squared minus b squared. Little bit of review. This is a times a, which is a squared. a times b, which is ab. Minus b times a, which is minus ab. And then, negative b times a positive b, negative b squared. These cancel out and you're just left with a squared minus b squared. So 2 minus the square root of 5 times 2 plus the square root of 5 is going to be equal to 2 squared, which is 4. Let me write it that way. It's going to be equal to 2 squared minus the square root of 5 squared, which is just 5. So this would just be equal to 4 minus 5 or negative 1. If you take advantage of the difference of squares of binomials, or the factoring difference of squares, however you want to view it, then you can rationalize this denominator. So let's do that. Let me rewrite the problem. 12 over 2 minus the square root of 5. In this situation, I just multiply the numerator and the denominator by 2 plus the square root of 5 over 2 plus the square root of 5. Once again, I'm just multiplying the number by 1. So I'm not changing the fundamental number. I'm just changing how we represent it. So the numerator is going to become 12 times 2, which is 24. Plus 12 times the square root of 5. Once again, this is like a factored difference of squares. This is going to be equal to 2 squared, which is going to be exactly equal to that. Which is 4 minus 1, or we could just-- sorry. 4 minus 5. It's 2 squared minus square root of 5 squared. So it's 4 minus 5. Or we could just write that as minus 1, or negative 1. Or we could put a 1 there and put a negative sign out in front. And then, no point in even putting a 1 in the denominator. We could just say that this is equal to negative 24 minus 12 square roots of 5. So this case, it kind of did simplify it as well. It wasn't just for the sake of rationalizing it. It actually made it look a little bit better. And you know, I don't if I mentioned in the beginning, this is good because it's not obvious. If you and I are both trying to build a rocket and you get this as your answer and I get this as my answer, this isn't obvious, at least to me just by looking at it, that they're the same number. But if we agree to always rationalize our denominators, we're like, oh great. We got the same number. Now we're ready to send our rocket to Mars. Let's do one more of this, one more of these right here. Let's do one with variables in it. So let's say we have 5y over 2 times the square root of y minus 5. So we're going to do this exact same process. We have a binomial with an irrational denominator. It might be a rational. We don't know what y is. But y can take on any value, so at points it's going to be irrational. So we really just don't want a radical in the denominator. So what is this going to be equal to? Well, let's just multiply the numerator and the denominator by 2 square roots of y plus 5 over 2 square roots of y plus 5. This is just 1. We are not changing the number, we're just multiplying it by 1. So let's start with the denominator. What is the denominator going to be equal to? The denominator is going to be equal to this squared. Once again, just a difference of squares. It's going to be 2 times the square root of y squared minus 5 squared. If you factor this, you would get 2 square roots of y plus 5 times 2 square roots of y minus 5. This is a difference of squares. And then our numerator is 5y times 2 square roots of y. So it would be 10. And this is y to the first power, this is y to the half power. We could write y square roots of y. 10y square roots of y. Or we could write this as y to the 3/2 power or 1 and 1/2 power, however you want to view it. And then finally, 5y times 5 is plus 25y. And we can simplify this further. So what is our denominator going to be equal to? We're going to have 2 squared, which is 4. Square root of y squared is y. 4y. And then minus 25. And our numerator over here is-- We could even write this. We could keep it exactly the way we've written it here. We could factor out a y. There's all sorts of things we could do it. But just to keep things simple, we could just leave that as 10. Let me just write it different. I could write that as this is y to the first, this is y to the 1/2 power. I could write that as even a y to the 3/2 if I want. I could write that as y to the 1 and 1/2 if I want. Or I could write that as 10y times the square root of y. All of those are equivalent. Plus 25y. Anyway, hopefully you found this rationalizing the denominator interesting.