# This note has been used to help create the Mental Math Tricks wiki

Hey guys, I saw a faster way to find cube roots.

We already know some basic cube numbers

\(0^{3}\)=0

\(1^{3}\)=1

\(2^{3}\)=8

\(3^{3}\)=27

\(4^{3}\)=64

\(5^{3}\)=125

\(6^{3}\)=216

\(7^{3}\)=343

\(8^{3}\)=512

\(9^{3}\)=729

Now, the common thing here is that each ones digit of the cube numbers is the same number that is getting cubed , except for 2 ,8 ,3 ,7 .

now let us take a cube no like 226981 .

to see which is the cube root of that number , first check the last 3 digits that is 981 . Its last digit is 1 so therefore the last digit of the cube root of 226981 is 1 .

Now for the remaining digits that is 226

Now 226 is the nearer & bigger number compared to the cube of 6 (216)

So the cube root of 226981 is 61

Let us take another example - 148877

Here 7 is in the last digit but the cube of seven's last digit is not seven. But the cube of three has the last digit as 7.

So the last digit of the cube root of 148877 is 3.

Now for the remaining digits 148.

It is the nearer and bigger than the cube of 5 (125).

Therefore the cube root of 148877 is 53.

Let us take another example 54872.

Here the last three digit's (872) last digit is 2 but the cube of 2's last digit is not 2 but the last of the cube of 8 is 2.

So the last digit of the cube root of 54872 is 8.

Now of the remaining numbers (54). It is nearer and bigger to the cube of 3 (27). So therefore the cube root of 54872 is 38.

## Comments

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TopNewestHow about to find cube roots of a number which answer is three-digit number ?? For example 111^3, 267^3, etc – Jonathan Christianto · 2 years, 6 months ago

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\(\sqrt[3]{1860867}\)

Done with the last three digits and the last digit , & you get 3 as the last digit of \(\sqrt[3]{1860867}\)

Now find the nearest cube number of1860 & it is 12 (1728)

So therefore \(\sqrt[3]{1860867}\) = 123 – Kartik Kulkarni · 2 years, 6 months ago

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– Jonathan Christianto · 2 years, 6 months ago

So we just do the same ways... Thank you so much..Log in to reply

– Patricio Ramos · 2 years, 5 months ago

waitwaitwaitwaitwait..whaaaaaaaaaaat? Where did that 3 even come from? The last digit of 1860867 is 7.....Log in to reply

– Kartik Kulkarni · 2 years, 5 months ago

Read the note properly , it says 7 is in the last digit but the cube of seven's last digit is not seven. But the cube of three has the last digit as 7.Log in to reply

– Patricio Ramos · 2 years, 5 months ago

So basically the cube of the number you are looking for must have the same last digit as the number in the problem?Log in to reply

1), 4 (4³ = 64), 5 (5³ = 125), 6 (6³ = 216), 9 (9³ = 729) and 0 (0³ =0). But, here we see, it doesn't happen for 2 (2³ =8), neither 3 (3³ = 27), nor 7 (7³ = 343) and 8 (8³ = 512). I'll always have to check this before find cube roots by this method. – Matheus Abrão Abdala · 2 years, 5 months agoLog in to reply

– Niranjan Khanderia · 2 years, 5 months ago

2, 3, and 7, 8 has at their unit place have their 10's compliments. Rest have the same number as said earlier.Log in to reply

Thank you, a great method to solve the cube roots, so bad it doesn't work with every cubic root, it would save a lot of time in tests. Anyway, thanks! – Guilherme Aleixo · 2 years, 5 months ago

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Another Interesting fact:: (A) cube of 2= unit digit 8 .....cube of 8=unit digit 2 (B) cube of 3=unit digit 7...... cube of 7= unit digit 3 – Bakul Majumder · 2 years, 5 months ago

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1, 4, 5, 6, 9 have the unit place of their cubes as the number themselves. But cubes of 2,3 and 8,7 has there unit place as their compliment of 10. – Niranjan Khanderia · 2 years, 5 months ago

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It works for groups of threes. How adorable. – Lovelli Fuad · 2 years, 5 months ago

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ecellent method .its working – Raj Miglani · 2 years, 5 months ago

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Nice method but I have a doubt. Let us take 117 649 as an example. The last digit is 9. Now 117 is nearer to 5^3(125) rather than 4^3(64). So why is the cube root of 117 649, 49 instead of 59. Would really appreciate an answer. – Samar Perwez · 5 hours ago

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I gave the first person who introduced this to me a very good comment. Features of natural numbers can occasionally been found. Important thing is never let this concept to mislead ourselves when the situation is not whole numbers. I recalled and remember ed again but not really memorized properly. Understand why could make me a better memory perhaps. Hope I can memorize from today onwards! – Lu Chee Ket · 2 months ago

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But it is not useful for non perfect cubes – Raj Bunsha · 2 months ago

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Write a comment or ask a question...if m=29 and e=13, then m=m+e e=m-e m=m-e then find the new value of m and e?? – Eyob Assefa · 1 year, 11 months ago

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Very nice & thanks. – Narendra Patki · 2 years, 3 months ago

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Good Method ....!!! Amazing...!! – Mohammad Dilshad · 2 years, 4 months ago

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thats just for a sure perfect cube – Angelo Forcadela · 2 years, 4 months ago

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Cool....... – Gaurav Negi · 2 years, 4 months ago

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Really very useful trick Thanks:) – R J · 2 years, 5 months ago

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It's really coolest method ever.but can any1 suggests me methods for square root of a decimal number.for eg:square root of 0.56 – Rushabh Shah · 2 years, 5 months ago

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how to work out cube root of 216216. The answer on face is 66 but that is not the cube root. – Sheikh Waseem · 2 years, 5 months ago

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– Anshul Gupta · 2 years, 5 months ago

you have to it by long division methodLog in to reply

Very nice and interesting solution – Istmio Veneroso · 2 years, 5 months ago

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good one – Abhijeet Verma · 2 years, 5 months ago

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I HAVE SOME CONFUSION THAT WHEN HAM LOG SAME NO. KO LIKHEGE OR KAB NHI........AS 1ST SUM MEN.......226 KA 6 LIKHE AND 981 KA 1 SO ANS. IS 61 BUT 148877 MEN 148 KA 5 KYU LAST NO TO 8 HA SO COMPLEMENTRY IS 2 BUT HERE IS 5.. – Mamta Ray · 2 years, 5 months ago

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– Kartik Kulkarni · 2 years, 5 months ago

I would prefer not to use Hindi cause it is confusing me that you have mixed up English & HindiLog in to reply

Only works for whole numbers. It's interesting however that you have found this method. How did you come across it? – Gui Lanham · 2 years, 5 months ago

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excellent method!!! upvoted young mind :) – Rohit Ner · 2 years, 5 months ago

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I like it – Ashish Gupta · 2 years, 5 months ago

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Who discovered this method? It's really awesome – Anshul Gupta · 2 years, 5 months ago

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i like that method – Kibria Robin · 2 years, 5 months ago

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Real nice method. I liked it. – Nurul Afsar · 2 years, 5 months ago

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I like this method . – Bakul Majumder · 2 years, 5 months ago

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nice – Ray Macedo · 2 years, 5 months ago

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you mean x^3 of 226981 , 226971 , 226961 , 226981 , 226881 , all is 61 only by your way. which is incorrect – Ashish Jaiswal · 2 years, 5 months ago

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– Tarunesh V · 2 years, 5 months ago

you have to know that it it works only for a perfect cubeLog in to reply

– Kartik Kulkarni · 2 years, 5 months ago

I'm sorry I did not understandLog in to reply

– Ashish Jaiswal · 2 years, 5 months ago

By this trick cube root for last 3 digit is depends on unit place digit only? if we consider these numbers which all have 1 as unit place digit , 226981 , 226971 , 226961 , 226221 , 226881 so by the rule cube root should be 61 for all these numbers. which is actually incorrect because numbers are different.Log in to reply

– Kartik Kulkarni · 2 years, 5 months ago

This method is only applicable for cube numbers that have the cube root with no numbers after the decimal pointLog in to reply

– Niranjan Khanderia · 2 years, 5 months ago

As I have mentioned in another comment, if the number is not a perfect cube, we at least know the floor and the ceiling of this number.Log in to reply

excellent – Venkata Kantipudi · 2 years, 5 months ago

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Thank U Very Much.I like Ur Way To solve The Problem. – Narendra Patki · 2 years, 5 months ago

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Great! Interesting! – Sheikh Waseem · 2 years, 5 months ago

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Very helpful. Thank you! – Arjun Manoj · 2 years, 5 months ago

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Really good method... I like it! – Mark Bray · 2 years, 5 months ago

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Good solution – Kuttiyam Srinivasan · 2 years, 5 months ago

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Write a comment or ask a question... Super – Chaitu Kvr · 2 years, 5 months ago

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Thanks – Qgc Gojra · 2 years, 5 months ago

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Excellent method – Diego Armando Pulido Ramos · 2 years, 5 months ago

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Just noticed. It actually isn't applicable to numbers other than perfect cubes. For example, if you calculate the cube root of 1,216 using this method, you get 16; actual root is 10.67. They're almost 5.5 numbers apart. If you have any better ways, please post it. – Yash Kapoor · 2 years, 5 months ago

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– Niranjan Khanderia · 2 years, 5 months ago

In that case we know between which two integers the actual cube root lays.Log in to reply

– Kartik Kulkarni · 2 years, 5 months ago

I had answered to a similar question , & this method is only applicable for numbers which have their cube roots with no numbers after the decimal pointLog in to reply

Awesome and unique way to do it!! Thanks!! – Yash Kapoor · 2 years, 5 months ago

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Nice note – Rifath Rahman · 2 years, 5 months ago

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Would largely help me for finding Karl Pearson's coefficient. Thanks. – Jay Mehta · 2 years, 6 months ago

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Fantastic method Thanks – Menna Attia · 2 years, 6 months ago

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Brilliant! Good to learn this from you. Thanks. – Lu Chee Ket · 2 years, 6 months ago

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maths is not about approximation and estimation!!!! – Sanket Kar · 2 years, 6 months ago

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can someone prove it mathematically? – Anirudh Roy · 2 years, 6 months ago

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– Anshul Gupta · 2 years, 5 months ago

https://brilliant.org/discussions/thread/long-divison-method-of-cube-root/Log in to reply

– Lu Chee Ket · 2 years, 6 months ago

a + 10 b + 100 c + 1000 d + 10000 e + 100000 f could roughly prove it I guess.Log in to reply

@Kartik Kulkarni .... really a nice one ... but i hav a doubt ... take 1331 ..... u get 11 by the method stated above .... if u take 1441 ..... 11 isnt correct ..... in that case .... u cant find whether a no. is a cube no. or not using this method.... rite??? – Ganesh Ayyappan · 2 years, 6 months ago

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– Kartik Kulkarni · 2 years, 6 months ago

also 1441's cube root is somewhat 11 And many more numbers after the decimal pointsLog in to reply

– Kartik Kulkarni · 2 years, 6 months ago

Well actually,this method is only applicable for actual cube numbersLog in to reply

– Anshul Gupta · 2 years, 5 months ago

Great buddy. Here is the actual method https://brilliant.org/discussions/thread/long-divison-method-of-cube-root/Log in to reply

@Kartik Kulkarni ... as soon as i saw ..... i found this interesting and also concluded this is applicable for perfect cubes ... but ur inference of 1441's cube root is around 11 is wrong ..... eg: take 1721 ..... if u infer by the same method as u did above ... it is around 11 ... but actually it can be estimated to 12 ..... (Note: cube root of 1721 = 11.98) – Ganesh Ayyappan · 2 years, 6 months ago

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– Kartik Kulkarni · 2 years, 6 months ago

well , I didn't think about the estimation partLog in to reply

cool – Vishwathiga Jayasankar · 2 years, 6 months ago

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Really cool way...I m looking forward to u to post some cool ways of finding the sum of series.... – Sarvesh Dubey · 2 years, 6 months ago

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according to this cube root of 125486 should be 56 but actually it is not – Devang Agrawal · 2 years, 3 months ago

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for eg 125486. last 3 digits = 6, first 3 digist =5,

here 125 is perfect cube , hence it doesnt work.

MY findings = This method only works when neither of the components( 1st 3 digits & last 3 digits) are perfect cube but the number that is comprised of the components is a perfect cube.

In ur case 125486 aint a perfect cube cum 125 which i call a component is.

Hows my Theorem? Thumbs up!! – Mohammed Ali · 2 years, 3 months ago

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– Niranjan Khanderia · 2 years, 3 months ago

OK. what about 125000?Log in to reply

125000, fr last 3 digits =0, fr first 3 digits =5

cube root of 125000 is 50.

(Notice that 000 is nothing but 0 and not 1000, 0^3 is 0) – Mohammed Ali · 2 years, 3 months ago

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– Sam Reeve · 1 year, 7 months ago

216216Log in to reply

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– Mohammed Ali · 2 years, 3 months ago

Sorry i didnt get u.Log in to reply

– Kartik Kulkarni · 2 years, 2 months ago

125486 is not a perfect cube and so this method is not applicable for that numberLog in to reply

knew that already – Qian Yu Hang · 2 years, 5 months ago

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do you want to know the exact long division method of finding cube roots though it tedious... :) – Anshul Gupta · 2 years, 5 months ago

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– Ailene Nunez · 2 years, 5 months ago

sure.Log in to reply

– Anshul Gupta · 2 years, 5 months ago

https://brilliant.org/discussions/thread/long-divison-method-of-cube-root/Log in to reply

whats's wrong with these four numbers(2,3,7 and 8)? i mean these are the number which you will never find at the end of any "squared number"( at ones place i mean). and here too the same four number have different digits at ones place. by the way nice trick. thanks! – Dhiraj Upadhyay · 2 years, 5 months ago

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what if we have 7 digited number could u explane me how to do it please – Sidharth Batchu · 2 years, 5 months ago

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– Kartik Kulkarni · 2 years, 5 months ago

I just explained it to Jonathan Christianto aboveLog in to reply

– Sidharth Batchu · 2 years, 5 months ago

kk:":":":":":":thankuLog in to reply

– Anshul Gupta · 2 years, 5 months ago

then we have to go by long divison actual methodLog in to reply