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Easy way to find cube roots

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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.

Note by Kartik Kulkarni
2 years, 2 months ago

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How 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, 2 months ago

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@Jonathan Christianto After doing the last three digits , try to find which is the nearest cube number to it for the remaining digits E.g -

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

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, 2 months ago

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@Kartik Kulkarni So we just do the same ways... Thank you so much.. Jonathan Christianto · 2 years, 2 months ago

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@Kartik Kulkarni waitwaitwaitwaitwait..whaaaaaaaaaaat? Where did that 3 even come from? The last digit of 1860867 is 7..... Patricio Ramos · 2 years, 1 month ago

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@Patricio Ramos 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. Kartik Kulkarni · 2 years, 1 month ago

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@Kartik Kulkarni So basically the cube of the number you are looking for must have the same last digit as the number in the problem? Patricio Ramos · 2 years, 1 month ago

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@Patricio Ramos No, dude. It occasionally happens, but it ain't no rule. It happens for 1 (1³ = 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, 1 month ago

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@Matheus Abrão Abdala 2, 3, and 7, 8 has at their unit place have their 10's compliments. Rest have the same number as said earlier. Niranjan Khanderia · 2 years, 1 month ago

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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, 1 month 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, 1 month 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, 1 month ago

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It works for groups of threes. How adorable. Lovelli Fuad · 2 years, 1 month ago

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ecellent method .its working Raj Miglani · 2 years, 1 month 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, 7 months ago

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Very nice & thanks. Narendra Patki · 1 year, 11 months ago

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Good Method ....!!! Amazing...!! Mohammad Dilshad · 2 years ago

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thats just for a sure perfect cube Angelo Forcadela · 2 years ago

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Cool....... Gaurav Negi · 2 years ago

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Really very useful trick Thanks:) R J · 2 years, 1 month 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, 1 month 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, 1 month ago

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@Sheikh Waseem you have to it by long division method Anshul Gupta · 2 years, 1 month ago

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Very nice and interesting solution Istmio Veneroso · 2 years, 1 month ago

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good one Abhijeet Verma · 2 years, 1 month 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, 1 month ago

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@Mamta Ray I would prefer not to use Hindi cause it is confusing me that you have mixed up English & Hindi Kartik Kulkarni · 2 years, 1 month ago

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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, 1 month ago

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excellent method!!! upvoted young mind :) Rohit Ner · 2 years, 1 month ago

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I like it Ashish Gupta · 2 years, 1 month ago

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Who discovered this method? It's really awesome Anshul Gupta · 2 years, 1 month ago

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i like that method Kibria Robin · 2 years, 1 month ago

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Real nice method. I liked it. Nurul Afsar · 2 years, 1 month ago

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I like this method . Bakul Majumder · 2 years, 1 month ago

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nice Ray Macedo · 2 years, 1 month 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, 1 month ago

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@Ashish Jaiswal you have to know that it it works only for a perfect cube Tarunesh V · 2 years, 1 month ago

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@Ashish Jaiswal I'm sorry I did not understand Kartik Kulkarni · 2 years, 1 month ago

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@Kartik Kulkarni 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. Ashish Jaiswal · 2 years, 1 month ago

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@Ashish Jaiswal This method is only applicable for cube numbers that have the cube root with no numbers after the decimal point Kartik Kulkarni · 2 years, 1 month ago

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@Kartik Kulkarni 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. Niranjan Khanderia · 2 years, 1 month ago

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excellent Venkata Kantipudi · 2 years, 1 month ago

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Thank U Very Much.I like Ur Way To solve The Problem. Narendra Patki · 2 years, 1 month ago

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Great! Interesting! Sheikh Waseem · 2 years, 1 month ago

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Very helpful. Thank you! Arjun Manoj · 2 years, 1 month ago

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Really good method... I like it! Mark Bray · 2 years, 1 month ago

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Good solution Kuttiyam Srinivasan · 2 years, 1 month ago

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Write a comment or ask a question... Super Chaitu Kvr · 2 years, 1 month ago

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Thanks Qgc Gojra · 2 years, 1 month ago

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Excellent method Diego Armando Pulido Ramos · 2 years, 1 month 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, 1 month ago

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@Yash Kapoor In that case we know between which two integers the actual cube root lays. Niranjan Khanderia · 2 years, 1 month ago

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@Yash Kapoor 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 point Kartik Kulkarni · 2 years, 1 month ago

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Awesome and unique way to do it!! Thanks!! Yash Kapoor · 2 years, 1 month ago

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Nice note Rifath Rahman · 2 years, 1 month ago

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

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

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

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

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

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@Anirudh Roy https://brilliant.org/discussions/thread/long-divison-method-of-cube-root/ Anshul Gupta · 2 years, 1 month ago

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@Anirudh Roy a + 10 b + 100 c + 1000 d + 10000 e + 100000 f could roughly prove it I guess. Lu Chee Ket · 2 years, 2 months ago

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@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, 2 months ago

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@Ganesh Ayyappan also 1441's cube root is somewhat 11 And many more numbers after the decimal points Kartik Kulkarni · 2 years, 2 months ago

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@Ganesh Ayyappan Well actually,this method is only applicable for actual cube numbers Kartik Kulkarni · 2 years, 2 months ago

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@Kartik Kulkarni Great buddy. Here is the actual method https://brilliant.org/discussions/thread/long-divison-method-of-cube-root/ Anshul Gupta · 2 years, 1 month ago

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@Kartik Kulkarni @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, 2 months ago

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@Ganesh Ayyappan well , I didn't think about the estimation part Kartik Kulkarni · 2 years, 2 months ago

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cool Vishwathiga Jayasankar · 2 years, 2 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, 2 months ago

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according to this cube root of 125486 should be 56 but actually it is not Devang Agrawal · 1 year, 12 months ago

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@Devang Agrawal I just found out when this method works,

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 · 1 year, 12 months ago

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@Mohammed Ali OK. what about 125000? Niranjan Khanderia · 1 year, 12 months ago

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@Niranjan Khanderia An adition to my findings: Either both the components aint perfect cube or both are.

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 · 1 year, 11 months ago

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@Mohammed Ali 216216 Sam Reeve · 1 year, 3 months ago

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Comment deleted Apr 05, 2015

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Comment deleted Apr 08, 2015

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@Niranjan Khanderia Sorry i didnt get u. Mohammed Ali · 1 year, 11 months ago

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@Devang Agrawal 125486 is not a perfect cube and so this method is not applicable for that number Kartik Kulkarni · 1 year, 10 months ago

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knew that already Qian Yu Hang · 2 years, 1 month 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, 1 month ago

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@Anshul Gupta sure. Ailene Nunez · 2 years, 1 month ago

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@Ailene Nunez https://brilliant.org/discussions/thread/long-divison-method-of-cube-root/ Anshul Gupta · 2 years, 1 month ago

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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, 1 month 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, 1 month ago

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@Sidharth Batchu I just explained it to Jonathan Christianto above Kartik Kulkarni · 2 years, 1 month ago

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@Kartik Kulkarni kk:":":":":":":thanku Sidharth Batchu · 2 years, 1 month ago

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@Sidharth Batchu then we have to go by long divison actual method Anshul Gupta · 2 years, 1 month ago

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