If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Cube root of a non-perfect cube

Created by Sal Khan.

Want to join the conversation?

  • leaf green style avatar for user Jake Rose
    I am so very lost.
    I have prime factorization down but this video leaves me utterly confused about how to find the cube root. The hints in the problems aren't very helpful. I do not understand how to go from prime factors to the cube root of non-perfect numbers. Please help!
    (24 votes)
    Default Khan Academy avatar avatar for user
    • starky tree style avatar for user Vaishakh
      You prime factorize the number then what you do is group the numbers appearing three times
      like 7 in above video.group the other numbers appearing in times which are not multiples of three (2 and 5 in above example). multiply them and write them in the form cuberoot of
      'multiplied number" (here 10 i.e 2*5).At last take the number you grouped first(number appearing three times) and write in form "X x cuberoot of y" where X=first grouped num..... y=second grouped num............ x=multiplied by
      (9 votes)
  • blobby green style avatar for user t.c.pearce
    I understand the method, but I spot two problems:

    * won't different people get different answers depending upon which factors they try first?
    * how do we know that the final surd form is the simplest?

    Or is this just an illustration of factoring?

    Thank you.
    (4 votes)
    Default Khan Academy avatar avatar for user
    • piceratops seedling style avatar for user bsd
      Well if you tried using different factors to start with, you will get the same answers eventually at the end when it is fully simplified. So in this example, you can start out by dividing by 7,5, or 49, for example, and the answer at the end for all three is still the same. You can tell when the final form is the simplest when all the numbers you have left are prime numbers or in other words they can't be broken down further (e.g. 2,3,5,7,11, etc.) Knowing what the prime numbers are comes with practice.
      (4 votes)
  • piceratops seed style avatar for user Ashley Miller
    What are imaginary numbers? How do they work?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • duskpin seedling style avatar for user @Amour.Laila
    What the easiest way to do a cube root with a non-perfect cube
    (3 votes)
    Default Khan Academy avatar avatar for user
  • leaf red style avatar for user Jack McClelland
    Are there other ways (other than prime factorization) to find the cube root of a number?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • leaf green style avatar for user ✌PIXholic Studios
    i am kinda lost please explain what sal means
    (3 votes)
    Default Khan Academy avatar avatar for user
  • piceratops seed style avatar for user Anthea OSEKU
    I was wondering, what method can one use to calculate the cube root of a prime number without a calculator? for example I am supposed to know how to calculate 2 to the power 1/3 without using a calculator.
    (2 votes)
    Default Khan Academy avatar avatar for user
  • piceratops ultimate style avatar for user d.eileen.d
    wait, i think i missed Exponent Properties... was that a video?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • female robot grace style avatar for user Liz
    At , Sal has to do long division to try to figure out the factors. Is there an easier/ less time consuming way of doing this? Like tricks for looking at numbers and being able to skip some parts of this lengthy process?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • sneak peak green style avatar for user Wendy
      Sal could have skipped the first long division, or 1715/5, by simply factoring out 10 first. This gives him factors of 10 and 343, easily seen without long division. The 10 then factors down into the same 2 and 5 he started with. For the 343, however, I don't know of an easy way to factor that further without the long division.
      (1 vote)
  • blobby green style avatar for user lobsang
    How to find cube root of a non cube prime number.Please help me
    (2 votes)
    Default Khan Academy avatar avatar for user

Video transcript

Let's see if we can find the cube root of 3,430. And if you're like me, it doesn't jump out of your mind what number times that same number times that same number-- if you have three of those numbers and you were to multiply them together-- would be equal to 3,430. So what I'm going to do is to try to prime factorize this to find all the prime factors of 3,430 and see if any of those prime factors show up at least three times. And that'll help us with this. So 3,430-- it's clearly divisible by 5 and 2, or it's divisible by 10. So let's do that. So first we can divide it by 2. It's 2 times-- let's see. 3,430 divided by 2 is 1,715. Then we can divide it by 5, as well. We can factor 1,715 into 5 and-- let me do a little bit of long division on the side here. So if I have 1,715, and I'm going to divide it by 5. 5 doesn't go into 1. It goes into 17 three times. 3 times 5 is 15. Subtract, you get 2, and then you bring down a 1. 5 goes into 21 four times. 4 times 5 is 20. Subtract. Bring down the 5. 5 goes into 15 three times, so it goes exactly 343 times. So 1,715 can be factored into 5 times 343. Now, 343 might not jump out at you as a number that is easy to factor. It's clearly an odd number, so it won't be divisible by 2. Its digits add up to 10, which is not divisible by 3. So this isn't going to be divisible by 3. It's not going to be divisible by 4, because it's not divisible by 2. It's not going to be divisible by 5. If it wasn't divisible by 3 or 2, it's not going to be divisible by 6. And now we get to 7. Usually when you see a nutty number like this that doesn't seem to be divisible by a lot of things, it's always a good idea to try things like 7, 11, 13. Because those tend to construct very interesting numbers. So let's see if this is divisible by 7. So if I take 343 and if I want to divide it by 7, 7 goes into 30-- it doesn't go into 3-- 7 goes into 34 four times. 4 times 7 is 28. Subtract, 34 minus 28 is 6. Bring down a 3. 7 goes into 63 nine times. 9 times 7 is 63. Subtract. We don't have any remainder. And I forgot to do that last step up here. 3 times 15 is 15. Subtract, no remainder. It went in exactly. So here, 343 can be factored into 7 and 49. And 49 might jump out at you. It can be factored into 7 times 7. So this is interesting. I can rewrite all of this here-- the cube root of 3,430-- now as the cube root of-- I'm just going to write it in its factored form-- 2 times 5 times-- I could write 7 times 7 times 7, or I could write times 7 to the third power. That captures these three 7's right over here. I have three 7's, and then I'm multiplying them together. So that's 7 to the third power. And from our exponent properties, we know that this is the exact same thing as the cube root of 2 times 5 times the cube root-- so let me do that in that same, just so we see what colors we're dealing with. So the cube root of 2 times 5, which is the cube root of 10, times the cube root-- and I think you see where this is going-- of 7 to the third power. Keeping track of the colors is the hard part. And the cube root of 10, we just leave it as 10. We know the prime factorization of 10 is 2 times 5, so you're not going to just get a very simple integer answer here. You would get some decimal answer here, but here you get a very clear integer answer. The cube root of 7 to the third, well, that's just going to be 7. So this is just going to be 7. So our entire thing simplifies. This is equal to 7 times the cube root of 10. And this is about as simplified as we can get just using hand arithmetic. If you want to get the exact number here, you're probably best off using a calculator.