Cube root 45,118,016 using abacus (Triple-root method 7)

[Set 45,118,016 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 3-digits case and 1 digit left shift occurs. 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 45,118,016
(Answer is 356)

"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.

45,118,016 -> (45|118|016) : 45 is the 1st group number. The root digits is 3.


Step 1: Set 45118016. First group is 45.


Step 2: Cube number smaller than 45 is 27=3^3. Place 3 on E as 1st root.


Step 3: Place 45-27=18 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: 18/9=2 remainder 0. Place 2 on G.


Step 7: Place remainder 00 on HI.


Step 8: 11/9=1 remainder 2.


Step 9: Place 1 on I.


Step 10: Place remainder 02 on JK.


Step 11: Divide 20 on GH by current root 3. 20/3=5 remainder 5.


Step 12: Place 5 on F as 2nd root.


Step 13: Place remainder 05 on GH.


Step 14: Subtract 2nd root^2 from 51 on HI. (-b^2)


Step 15: Place 51-5^2=26 on HI.


Step 16: Multiply triple root 9 by remainder 26 on HI. 9X26=234


Step 17: Replace 26 by 00 on HI.


Step 18: Add 234 to 002 on IJK.


Step 19: It means place 002+234=236 on IJK.


Step 20: Subtract 2nd root^3 from 368 on JKL. (-b^3)


Step 21: It means place 368-5^3=243 on JKL.


Step 22: Focus on triple root (ABC).


Step 23: Add 3x2nd root to triple root root on BC. It means place 90+3x5=105 on ABC.


Step 24: As triple root carried 1 digit, it means 1st root becomes more than 4 digits form.


Step 25: Then, shift current root 35 to 1 digit left.


Step 26: Repeat division by triple root 105 until fixed position. (÷3a)


Step 27: Divide 224 on IJK by triple root 105. Place 224/105=2 remainder 14. Place 2 on G.


Step 28: Place 014 on IJK.


Step 29: 143/105=1 remainder 38


Step 30: Place 1 on H.


Step 31: Place remainder 038 on JKL.


Step 32: 380/105=3 remainder 65


Step 33: Place 3 on I.


Step 34: Place remainder 065 on KLM.


Step 35: 651/105=6 remainder 21


Step 36: Place 6 on J.


Step 37: Place remainder 021 on LMN.


Step 38: Divide 213 by current root 35.


Step 39: 213/35=6 remainder 3. Place 6 on F as 3rd root.


Step 40: Place remainder 003 on GHI.


Step 41: Subtract 3rd root^2 from 36 on IJ. (-c^2)


Step 42: Place 36-6^2=00 on IJ.


Step 43: Subtract 3rd root^3 from 216 on MNO. (-c^3)


Step 44: Subtract 3rd root^3 from 216 on MNO.


Step 45: Cube root of 45118016 is 356.


Final state: Answer 356

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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