Cube root 59,319 using abacus (Triple-root method 4)

[Set 59,319 on Mr. Cube root]

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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 2-digits case and we require 9 as root in the steps. 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 59,319
(Answer is 39)

"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.

59,319 -> (59|319) : 59 is the 1st group number. The root digits is 2.


Step 1: Set 59319. First group is 59.


Step 2: Cube number smaller than 59 is 27=3^3. Place 3 on E as 1st root.


Step 3: Place 59-27=32 on HI. (-a^3)


Step 4: Place Triple root 3x3=9 on B.


Step 5: Repeat division by triple root 9 until 4th digits next to 1st root. (÷3a)


Step 6: 32/9=3 remainder 5. Place 3 on G.


Step 7: Place remainder 05 on HI.


Step 8: 53/9=5 remainder 8. Place 5 on H.


Step 9: Place remainder 08 on IJ.


Step 10: 81/9=9 remainder 0.


Step 11: Place remainder 00 on JK.


Step 12: As the last digit of triple root B equals to the last digit of remaining root I, we set 9 on F as 2nd root.


Step 13: 35/9=3 remainder 8. (-b^2)


Step 14: Place remainder 08 on GH.


Step 15: Focus on 89.


Step 16: Place 89-2nd root^2=89-9^2=08 on HI.


Step 17: Multiply triple root B by remainder of the square subtraction 08. Place 08x9=72 on JK.


Step 18: Subtract 2nd root^3 from 729. (-b^3)


Step 19: Place 729-9^3=000 on JKL.


Step 20: Cube root of 59319 is 39.


Final state: Answer 39

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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