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Cube root 5,735,339 using abacus (Triple-root method 9)

2017年08月05日 21時44分35秒 | 開平、開立
[Set 5,735,339 on Mr. Cube root]


[Japanese]

Today's example is also about actual solution of Cube root using abacus. The calculation becomes more complicated than previous example.

Today's example is simple - basic Triple-root method, root is 3-digits case and requires excessive root modification. Please check the Theory page for your reference.

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 5,735,339
(Answer is 179)


"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.

5,735,339 -> (5|735|339): 5 is the 1st group number. The root digits is 3.


Step 1: Set 5735339. First group is 5.


Step 2: Cube number smaller than 5 is 1=1^3. Place 1 on E as 1st root.


Step 3: Place 5-1=4 on I. ( -a^3)


Step 4: Place Triple root 3x1=3 on B.


Step 5: Repeat division by triple root 3 until 4th digits next to 1st root. (/3a)


Step 6: 4/3=1 remainder 1. Place 1 on G.


Step 7: Place remainder 1 on I.


Step 8: 17/3=5 remainder 2.


Step 9: Place 5 on H.


Step 10: Place remainder 02 on IJ.


Step 11: 23/3=7 remainder 2.


Step 12: Place 7 on I.


Step 13: Place remainder 02 on JK.


Step 14: Place 8 on F as 2nd root according to the calculation rule.


Step 15: Divide 15 on GH by 2nd root. 15/8=1 remainder 7.


Step 16: Place 07 on GH.


Step 17: Subtract 2nd root^2 from 77 on HI. ( -b^2)


Step 18: It means place 77-8^2=13 on HI.


Step 19: Multiply triple root=3 by remainder 13 on HI. 3X13=39


Step 20: Set 00 on HI.


Step 21: Add 39 to 0002 on HIJK. 39+2=41


Step 22: It means place 0041 on HIJK.


Step 23: Cannot subtract 8^3 from 415 on JKL. 8 is excessive root. Subtract 1 from 8. Place 7 on F.


Step 24: Give back 10+8+7=25 on HI.


Step 25: Multiply triple root=3 by remainder 25 on HI. 3X25=75


Step 26: Set 00 on HI.


Step 27: Add 75 to 041 on IJK.


Step 28: It means place 041+75=116 on IJK.


Step 29: Subtract 2nd root^3 from 1165 on IJKL. 1165-7^3=822 ( -b^3)


Step 30: Place 0822 on IJKL.


Step 31: Add 3x2nd root to triple root 30. Place 30+21=51 on BC.


Step 32: Repeat division by triple root 51 from J.


Step 33: 82/51=1 remainder 31. Place 1 on H. Place remainder 31 on JK.


Step 34: 312/51=6 remainder 6. Place 6 on I.


Step 35: Place remainder 006 on JKL.


Step 36: 63/51=1 remainder 12. Place 1 on J.


Step 37: Place remainder 12 on LM.


Step 38: 123/51=2 remainder 21. Place 2 on K.


Step 39: Place remainder 021 on LMN.


Step 40: Divide 161 by current root 17. 161/17=9 remainder 8.


Step 41: Place 9 on G as 3rd root.


Step 42: Place remainder 008 on HIJ.


Step 43: Subtract 3rd root^2 from 82. 82-9^2=1 ( -c^2)


Step 44: Place 01 on JK.


Step 45: Multiply triple root=51 by remainder 01 on JK. 51X1=51


Step 46: Set 00 on JK.


Step 47: Add 51 to 21 on MN. 21+51=72.


Step 48: Place 72 on MN.


Step 49: Subtract 3rd root^3 from 729 on MNO. ( -c^3)


Step 50: Place 729-9^3=000 on MNO.


Step 51: Cube root of 5735339 is 179.


Final state: Answer 179

Abacus state transition. (Click to Zoom)




Next article is also about Cube root calculation (Triple-root method).


Related articles:

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

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


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