How to Calculate a Cube Root Using the Division Method

One of my viewers has asked me to produce a video showing how to calculate cube roots using the "division method." Thank you, Prashant. This video is for you.

In the firt part of the video I demonstrate the method that I prefer (there are "variations on the theme"). I then explain how the method works by discussing the binomial expansion of
(a + b)³ = a³ + 3a²b + 3ab² + b³.

If you have watched my earlier videos, you will see that the "theory" behind the method is essentially the same as that which lies behind the iterative method(s).

Because calculators are so readily available now, this skill is no longer used in any practical sense. Historically, it was one of the methods used to calculate roots for the mathematical tables that were used before mechanical calculators were easy to obtain.

The reason for my presenting it is to help better students gain insights into how algebra, graphing and calculus can help us understand how to manipulate numbers. So much theory of numbers is developed this way that this is a very worthwhile video for the serious mathematics student.

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Best wishes for your study and your mathematics!

[douglasfurman]

Dear Sir,
Thank you for publishing this video on a cube root algorithm. I appreciate that you showed the algebraic justification too. I appreciate learning about historical mathematics.
Best regards,
Doug

[dennisbell]

Graeme,
I reviewed your presentation of calculating cube roots and now understand the method well enough that I even calculated the fifth root of a couple of rather large numbers.As you noted, the divisors and remainders do get rather large as you proceed through the method, making it not very practical above finding say a cube root.However, it is a lost skill and I'm very glad that you responded to that request and demostrated it for those of us who do appreciate the " old school " methods.Many moons ago when I was in grade school, I remember doing by rote the mechanical method of calculating square roots.Thanks to you, now it is finally crystal clear.

[cipherunity]

This is my first time to watch your presentation. I must say it is too good.

[awaviarybangalore]

I actually love maths but nowhere i found very useful content that teaches me about cubic roots, but when i switched to youtube all were about solving perfect cubes and i needed imperfect ones. So salute to you sir from India.👌

[ishitarathii]

Couldn't find such an excellent explanation anywhere in school or youtube until i watched your video sir.... thank you so very much... i have an exam and this is very helpful.... recomending this video to everyone who are confused in finding the cube roots numbers

[ronalddump]

Btw, I am STILL the only person on earth (that I know of) who has demonstrated a practical method to find the cuberoot of any number using the long division method! Someone prove me wrong!

[gregorynelson]

I was playing around with the square root algorithm and sort of brute forced my way to this cube root algorithm by looking for patterns and testing, but I didn't know why it worked. It was brilliance when you showed and explained why it worked, that made so much sense now and I understand why the square root algorithm works now too. Thank you for this video.

[michaelknight]

Wow. This is such a great video. I love your explanation at the end showing how the process relates to (a+b)^3. Great work!

[rajdeepbakolia]

was having difficulty in my chemistry exam for finding cube root, you helped a lot sir thank you:)

[WorldVibes]

That was an excellent video to explain what cube root is all about. You explained not only the way to calculate the cube root but also showed why that way is correct. I have seen videos where shortcuts are explained but nobody knows why they are correct. Thanks for making the concepts so clear to students watching this video.

[Meghalova]

Thank you so very much for putting up this video. It really helped me to learn the long division method. Thank you so much.

[gerulais]

I loved the binomial presentation. Until then I couldn't understand the whole multiplication with 30, 300 and so on. Thank you, sir.

[TranquilSeaOfMath]

Thank you for posting this. I like how your algebraic explanation ties into the second course in algebra from the Ohio (U.S.A.) education standards. Best wishes. Cheerful Calculations!

[forrestgumpAMORFATI]

It's "Crystal Clear" sir thx alot!

[miguelreynel]

¡Excelente! Now I understand why, when extracting square roots, you have to multiply by 2 prior to look for the next number in the root, it’s the same logic. Thanks.

[dennisbell]

I'm pressed for time, but would like to thank you for the demostrating the procedure for calculating the cube root of a number using an " old school " method.I'm going to spend the time practicing on some additional examples and will get back with you additional comments.I, too, enjoy the history of mathematics and the individuals who created it.Thanks again.

[gabrielgarcia]

I have not been able to find the estimate vs difference shown like this anywhere else. Well done.

[DarthVader]

Nice Work. I was looking for a video like this which could algebraically explain the algorithm of calculating the roots of a rational number. Thank You. 😁

[martinryan]

Hi Sir, I am so pleased to have found this video which is the best explanation that I have come across after 3 days of searching on the internet. I have an old maths book which was handed down by my grand-aunts who were teachers in the early part of the 20th century. The book does not show when it was published but I suspect it was around the 1900s. There are a couple of examples of how to do the long calculation but I did not manage to follow the logic. I learned how to do cube roots in the very early 1960s (before calculators!) and was good at maths in general.I am now 70 and need to keep the remaining few brain cells ticking over....don't ask me why. By the way, I had difficulty with trigonometry when at school but managed to learn a quite a bit of it in the past year. If I keep going I might even sit for a degree, only joking. I notice from one of your replies that you were having some medical issue so I hope you are now back to full health. Just in case you want to check up on the maths book; it was published by G.H. Bradford, London under series titled ' Jack's Mathematical Series, Intermediate Arithmetic. Regards and Thank You, martinjr.

[jimcurrie]

I like your method. My high school teacher was a little bit more abstract on cube roots