Cube root 421,875 using abacus (1/3-multiplication table method 2)

[Set 421,875 on Mr. Cube root]

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Following the last time, today's example is about actual solution of Cube root using abacus.

Today's example is simple - basic 1/3-multiplication table method, root is 2-digits case. You can check the Index page of all articles.

Cube root methods: Triple-root method, constant number method, 3a^2 method, 1/3-division method, 1/3-multiplication table method, 1/3-multiplication table alternative method, Multiplication-Subtraction method, 3-root^2 method, Mixing method, Exceed number method, Omission Method, etc.


Abacus steps to solve Cube root of 421,875
(Answer is 75)

"1st group number" is the left most numbers in the 3-digits groups of the given number for cube root calculation. Number of groups is the number of digits of the Cube root.

421,875 -> (421|875): 421 is the 1st group number. The root digits is 2.


Step 1: Place 421875 on GHIJKL.


Step 2: The 1st group is 421.


Step 3: Cube number ≦ 421 is 343=7^3. Place 7 on C as the 1st root.


Step 4: Subtract 7^3 from the 1st group 421. Place 421-7^3=078 on GHI.


Step 5: Focus on 78875 on HIJKL.


Step 6: Divide 78875 by 3. Place 78875/3=26291.6 on HIJKLM.


Step 7: Focus on 26 on HI.


Step 8: Repeat division by triple root 7 until 4th digits next to 1st root. 26/7=3 remainder 5. Place 3 on E.


Step 9: Place remainder 05 on HI.


Step 10: Divide 52 on IJ by current root 7. 52/7=7 remainder 3


Step 11: Place 7 on F.


Step 12: Place 03 on IJ.


Step 13: Divide 39 on JK by current root 7. 39/7=5 remainder 4


Step 14: Place 5 on G.


Step 15: Place 04 on JK.


Step 16: Divide 37 on EF by current root 7. 37/7=5 remainder 2


Step 17: Place 5 on D as 2nd root.


Step 18: Place 02 on EF.


Step 19: Subtract 2nd root^2 from 25 on FG. 25-5^2=0


Step 20: Place 00 on FG.


Step 21: 00 on FG x 1st root and add 04.1 on JKL. 0X7+4.1=4.1


Step 22: Place 04.1 on JKL. It means do nothing.


Step 23: Subtract 2nd root^3/3 from 41.6 on KLM. 41.6-5^3/3=0


Step 24: Place 00.0 on KLM.


Step 25: Cube root of 421875 is 75.


Final state: Answer 75

Abacus state transition. (Click to Zoom)


It is interesting to compare with the Triple-root method.


Next article is also 1/3-multiplication table method.


Related articles:

How to solve Cube root of 1729.03 using abacus? (Feynman v.s. Abacus man)
https://blog.goo.ne.jp/ktonegaw/e/cff5d6e7ecaa07230b9cc7af10b23aed

Index: Square root and Cube root using Abacus
https://blog.goo.ne.jp/ktonegaw/e/f62fb31b6a3a0417ec5d33591249451b

Cube root 421,875 using abacus (Triple-root method 1)
https://blog.goo.ne.jp/ktonegaw/e/6a4c5516b2ede10271ac7e718daa4815


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