Cube root 110,592 using abacus (1/3-multiplication table method 3)

[Set 110,592 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 and we require root reduction in the steps. 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 110,592
(Answer is 48)

"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.

110,592 -> (110|592): 110 is the 1st group number. The root digits is 2.


Step 1: Place 110592 on GHIJKL.


Step 2: The 1st group is 110.


Step 3: Cube number ≦ 110 is 64=4^3. Place 4 on C as the 1st root.


Step 4: Subtract 4^3 from the 1st group 110. Place 110-4^3=046 on GHI.


Step 5: Focus on 46592 on HIJKL.


Step 6: Divide 46592 by 3. Place 46592/3=15530.6 on HIJKLM.


Step 7: Focus on 15 on HI.


Step 8: Repeat division by triple root 4 until 4th digits next to 1st root. 15/4=3 remainder 3. Place 3 on E.


Step 9: Place remainder 03 on HI.


Step 10: Divide 35 on IJ by current root 4. 35/4=8 remainder 3


Step 11: Place 8 on F.


Step 12: Place 03 on IJ.


Step 13: Divide 33 on JK by current root 4. 33/4=8 remainder 1


Step 14: Place 8 on G.


Step 15: Place 01 on JK.


Step 16: Divide 38 on EF by current root 4. 38/4=9 remainder 2


Step 17: Place 9 on D as 2nd root. (Temporary root)


Step 18: Place 02 on EF.


Step 19: Cannot subtract 9^2=81 from 28. Temporary root 9 is excessive root.


Step 20: Subtract 1 from excessive root 9. Place 8 on D.


Step 21: Return current root 2 on E. Place 2+4=6 on F.


Step 22: Place 68-2nd root^2=68-8^2=04 on FG.


Step 23: 04 on FG x 1st root 4 + 01 on JK. Place 4x4+1=17 on JK.


Step 24: Subtract 2nd root 8^3/3 from 176 on KLM. 176-8^3/3=0


Step 25: Place 000 on KLM.


Step 26: Cube root of 110592 is 48.


Final state: Answer 48

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 110,592 using abacus (Triple-root method 3)
https://blog.goo.ne.jp/ktonegaw/e/ca623f02e6f5fdfef4221a5c43ea1a95


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