Cube root 421,875 using abacus (Triple-root method 1)

[Set 421,875 on Mr. Cube root]

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We start the actual solutions to calculate Cube root using abacus. The calculation becomes more complicated but interesting.

Today's example is simple - basic Triple-root method, root is 2-digits case and 1st root is greater than or equal to 4.. 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 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: Set 421875. First group is 421.


Step 2: Cube number smaller than 421 is 343=7^3. Place 7 on D as 1st root.


Step 3: Place 421-343=078 on GHI. (-a^3)


Step 4: Place Triple root 3x7=21 on AB.


Step 5: Repeat division by 21 until 4th digits next to 1st root. (/3a)


Step 6: 78/21=3 remainder 15. Place 3 on F and 15 on HI.


Step 7: Divide 158 by 21.


Step 8: 158/21=7 remainder 11. Place 7 on G and 011 on HIJ.


Step 9: Divide 117 by 21.


Step 10: 117/21=5 remainder 12 Place 5 on H and 012 on IJK.


Step 11: Divide 37 by current root (1st root) 7. (/a)


Step 12: Ansewer is 5 and place 5 on E as 2nd root.


Step 13: Place 37-1st root x Answer=37-7x5=02 on FG. (-ab)


Step 14: Focus on 25 on GH.


Step 15: Place 25-2nd root^2=25-5^2=00 on GH. (-b^2)


Step 16: Focus on 125 on JKL.


Step 17: Place 125-2nd root^2=125-5^3=000 on JKL. (-b^3)


Step 18: Cube root is 75.


Final state: Answer 75

Abacus state transition. (Click to Zoom)


It is interesting to compare with the 1/3-multiplication table method.


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

Cube root 421,875 using abacus (1/3-multiplication table method 2)
http://blog.goo.ne.jp/ktonegaw/e/502f66297303cb36356f61ac6de3936b


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