[Set 12,895,213,625 on Mr. Cube root]Zoom
[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 4-digits case. 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 12,895,213,625
(Answer is 2,345)
"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.
12,895,213,625 -> (12|895|213|625): 12 is the 1st group number. The root digits is 4.
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Step 1: Set 12895213625. First group is 12.
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Step 2: Cube number smaller than 12 is 8=2^3. Place 2 on E as 1st root.
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Step 3: Place 12-8=04 on HI. ( -a^3)
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Step 4: Place Triple root 3x2=6 on B.
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Step 5: Repeat division by triple root 6 until 4th digits next to 1st root. ( /3a)
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Step 6: 48/6=8 remainder 0. Place 8 on G.
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Step 7: Place remainder 00 on IJ.
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Step 8: 9/6=1 remainder 3.
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Step 9: Place 1 on I.
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Step 10: Place remainder 3 on K.
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Step 11: Divide 0 on F by current root 2.
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Step 12: Get 3 as 2nd root and place it on F.
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Step 13: Place 8-2x3=2 on H.
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Step 14: Subtract 2nd root^2 from 21 on HI. ( -b^2)
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Step 15: Place 21-3^2=12 on HI.
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Step 16: Multiply triple root 6 by remainder 12 on HI. 6X12=72
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Step 17: Replace 12 by 00 on HI.
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Step 18: Add 72 to 03 on JK.
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Step 19: It means place 03+72=75 on JK.
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Step 20: Subtract 2nd root^3 from 755 on JKL. ( -b^3)
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Step 21: It means place 755-3^3=728 on JKL.
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Step 22: Add 3x2nd root to triple root on BC.
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Step 23: Place 60+3x3=69 on BC.
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Step 24: Repeat division by triple root 69 until fixed position.
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Step 25: 72/69=1 remainder 3. Place 1 on H.
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Step 26: Place remainder 03 on JK.
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Step 27: 38/69=0 remainder 38
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Step 28: Place 0 on I. Place remainder 38 on KL.
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Step 29: 382/69=5 remainder 37
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Step 30: Place 5 on J.
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Step 31: Place remainder 037 on JKL.
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Step 32: 371/69=5 remainder 26
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Step 33: Place 5 on K.
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Step 34: Place remainder 026 on LMN.
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Step 35: Divide 105 on HIJ by current root 23. 105/23=4 remainder 13
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Step 36: Place 4 on G as 3rd root.
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Step 37: Place remainder 013 on HIJ.
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Step 38: Subtract 3rd root^2 from 135 on IJK. ( -c^2)
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Step 39: Place 135-4^2=119 on IJK.
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Step 40: Multiply triple root 69 by remainder 119 on IJK. 69X119=8211
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Step 41: Replace 119 by 000 on IJK.
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Step 42: Add 8211 to 0026 on KLMN.
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Step 43: It means place 0026+8211=8237 on KLMN.
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Step 44: Subtract 3rd root^3 from 73 on NO. ( -c^3)
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Step 45: Place 73-4^3=09 onNO.
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Step 46: Add 3x3rd root to triple root on BCD.
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Step 47: Place 690+3x4=702 on BCD.
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Step 48: Repeat division by triple root 702 until fixed position.
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Step 49: 823/702=1 remainder 121. Place 1 on I.
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Step 50: Place remainder 121 on KLM.
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Step 51: 1210/702=1 remainder 508
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Step 52: Place 1 on J.
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Step 53: Place remainder 0508 on KLMN.
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Step 54: 5089/702=7 remainder 175
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Step 55: Place 7 on K.
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Step 56: Place remainder 0175 on LMNO.
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Step 57: 1756/702=2 remainder 352
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Step 58: Place 2 on L.
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Step 59: Place remainder 0352 on MNOP.
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Step 60: 3522/702=5 remainder 12
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Step 61: Place 5 on M.
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Step 62: Place remainder 0012 on NOPQ.
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Step 63: Divide 1172 on JKLM by current root 234. 1172/234=5 remainder 2
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Step 64: Place 5 on G as 4th root.
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Step 65: Place remainder 0002 on IJKL.
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Step 66: Subtract 4th root^2 from 25 on LM. ( -d^2)
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Step 67: Place 25-5^2=00 on LM.
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Step 68: Subtract 4th root^3 from 125 on PQR. (-d^2)
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Step 69: Place 125-5^3=000 on PQR.
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Step 70: Cube root of 12895213625 is 2345.
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Final state: Answer 2345
Abacus state transition. (Click to Zoom)
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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
Please place your mouse on the buttons and click one by one. These are blog ranking sites.
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[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 4-digits case. 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 12,895,213,625
(Answer is 2,345)
"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.
12,895,213,625 -> (12|895|213|625): 12 is the 1st group number. The root digits is 4.
Step 1: Set 12895213625. First group is 12.
Step 2: Cube number smaller than 12 is 8=2^3. Place 2 on E as 1st root.
Step 3: Place 12-8=04 on HI. ( -a^3)
Step 4: Place Triple root 3x2=6 on B.
Step 5: Repeat division by triple root 6 until 4th digits next to 1st root. ( /3a)
Step 6: 48/6=8 remainder 0. Place 8 on G.
Step 7: Place remainder 00 on IJ.
Step 8: 9/6=1 remainder 3.
Step 9: Place 1 on I.
Step 10: Place remainder 3 on K.
Step 11: Divide 0 on F by current root 2.
Step 12: Get 3 as 2nd root and place it on F.
Step 13: Place 8-2x3=2 on H.
Step 14: Subtract 2nd root^2 from 21 on HI. ( -b^2)
Step 15: Place 21-3^2=12 on HI.
Step 16: Multiply triple root 6 by remainder 12 on HI. 6X12=72
Step 17: Replace 12 by 00 on HI.
Step 18: Add 72 to 03 on JK.
Step 19: It means place 03+72=75 on JK.
Step 20: Subtract 2nd root^3 from 755 on JKL. ( -b^3)
Step 21: It means place 755-3^3=728 on JKL.
Step 22: Add 3x2nd root to triple root on BC.
Step 23: Place 60+3x3=69 on BC.
Step 24: Repeat division by triple root 69 until fixed position.
Step 25: 72/69=1 remainder 3. Place 1 on H.
Step 26: Place remainder 03 on JK.
Step 27: 38/69=0 remainder 38
Step 28: Place 0 on I. Place remainder 38 on KL.
Step 29: 382/69=5 remainder 37
Step 30: Place 5 on J.
Step 31: Place remainder 037 on JKL.
Step 32: 371/69=5 remainder 26
Step 33: Place 5 on K.
Step 34: Place remainder 026 on LMN.
Step 35: Divide 105 on HIJ by current root 23. 105/23=4 remainder 13
Step 36: Place 4 on G as 3rd root.
Step 37: Place remainder 013 on HIJ.
Step 38: Subtract 3rd root^2 from 135 on IJK. ( -c^2)
Step 39: Place 135-4^2=119 on IJK.
Step 40: Multiply triple root 69 by remainder 119 on IJK. 69X119=8211
Step 41: Replace 119 by 000 on IJK.
Step 42: Add 8211 to 0026 on KLMN.
Step 43: It means place 0026+8211=8237 on KLMN.
Step 44: Subtract 3rd root^3 from 73 on NO. ( -c^3)
Step 45: Place 73-4^3=09 onNO.
Step 46: Add 3x3rd root to triple root on BCD.
Step 47: Place 690+3x4=702 on BCD.
Step 48: Repeat division by triple root 702 until fixed position.
Step 49: 823/702=1 remainder 121. Place 1 on I.
Step 50: Place remainder 121 on KLM.
Step 51: 1210/702=1 remainder 508
Step 52: Place 1 on J.
Step 53: Place remainder 0508 on KLMN.
Step 54: 5089/702=7 remainder 175
Step 55: Place 7 on K.
Step 56: Place remainder 0175 on LMNO.
Step 57: 1756/702=2 remainder 352
Step 58: Place 2 on L.
Step 59: Place remainder 0352 on MNOP.
Step 60: 3522/702=5 remainder 12
Step 61: Place 5 on M.
Step 62: Place remainder 0012 on NOPQ.
Step 63: Divide 1172 on JKLM by current root 234. 1172/234=5 remainder 2
Step 64: Place 5 on G as 4th root.
Step 65: Place remainder 0002 on IJKL.
Step 66: Subtract 4th root^2 from 25 on LM. ( -d^2)
Step 67: Place 25-5^2=00 on LM.
Step 68: Subtract 4th root^3 from 125 on PQR. (-d^2)
Step 69: Place 125-5^3=000 on PQR.
Step 70: Cube root of 12895213625 is 2345.
Final state: Answer 2345
Abacus state transition. (Click to Zoom)
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
Please place your mouse on the buttons and click one by one. These are blog ranking sites.
