Cube root 721,734,273 using abacus (Triple-root method 8)

[Set 721,734,273 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 3-digits case and requires 9 as root in the middle of calculation. 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 Square root of 721,734,273
(Answer is 897)

"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.

721,734,273 -> (721|734|273): 721 is the 1st group number. The root digits is 3.

Image may be NSFW.
Clik here to view.
Step 1: Set 721734273. First group is 721.

Image may be NSFW.
Clik here to view.
Step 2: Cube number smaller than 721 is 512=8^3. Place 8 on D as 1st root.

Image may be NSFW.
Clik here to view.
Step 3: Place 721-512=209 on GHI.(-a^3)

Image may be NSFW.
Clik here to view.
Step 4: Place Triple root 3x8=24 on AB.

Image may be NSFW.
Clik here to view.
Step 5: Repeat division by triple root 24 until 4th digits next to 1st root.(/3a)

Image may be NSFW.
Clik here to view.
Step 6: 209/24=8 remainder 17. Place 8 on F.

Image may be NSFW.
Clik here to view.
Step 7: Place remainder 017 on GHI.

Image may be NSFW.
Clik here to view.
Step 8: 117/24=7 remainder 9.

Image may be NSFW.
Clik here to view.
Step 9: Place 7 on G.

Image may be NSFW.
Clik here to view.
Step 10: Place remainder 009 on HIJ.

Image may be NSFW.
Clik here to view.
Step 11: 93/24=3 remainder 21.

Image may be NSFW.
Clik here to view.
Step 12: Place 3 on H.

Image may be NSFW.
Clik here to view.
Step 13: Place remainder 21 on JK.

Image may be NSFW.
Clik here to view.
Step 14: Divide 87 on FG by current root 8. 87/8=10 remainder 7.

Image may be NSFW.
Clik here to view.
Step 15: Place 9 as 2nd root on E according to the calculation rule.

Image may be NSFW.
Clik here to view.
Step 16: Divide 87 on FG by current root 9. 87/9=8 remainder 15.

Image may be NSFW.
Clik here to view.
Step 17: Place remainder 15 on FG.

Image may be NSFW.
Clik here to view.
Step 18: Subtract 2nd root^2 from 153 on FGH. (-b^2)

Image may be NSFW.
Clik here to view.
Step 19: Place 153-9^2=072 on FGH.

Image may be NSFW.
Clik here to view.
Step 20: Multiply triple root 24 by remainder 72 on GH. 24X72=1728

Image may be NSFW.
Clik here to view.
Step 21: Replace 72 by 00 on GH.

Image may be NSFW.
Clik here to view.
Step 22: Add 1728 to 0021 on HIJK.

Image may be NSFW.
Clik here to view.
Step 23: It means place 0021+1728=1749 on HIJK.

Image may be NSFW.
Clik here to view.
Step 24: Subtract 2nd root^3 from 7494 on IJKL. (-b^3)

Image may be NSFW.
Clik here to view.
Step 25: It means place 7494-9^3=6765 on IJKL.

Image may be NSFW.
Clik here to view.
Step 26: Focus on triple root (ABC).

Image may be NSFW.
Clik here to view.
Step 27: Add 3x2nd root to triple root root on BC. Place 240+3x9=267 on ABC.

Image may be NSFW.
Clik here to view.
Step 28: Repeat division by triple root 267 until fixed position. (/3a)

Image may be NSFW.
Clik here to view.
Step 29: Divide 1676 on HIJK by triple root 267. Place 1676/267=6 remainder 74. Place 6 on G.

Image may be NSFW.
Clik here to view.
Step 30: Place 0074 on HIJK.

Image may be NSFW.
Clik here to view.
Step 31: 745/267=2 remainder 211

Image may be NSFW.
Clik here to view.
Step 32: Place 2 on H.

Image may be NSFW.
Clik here to view.
Step 33: Place remainder 211 on JKL.

Image may be NSFW.
Clik here to view.
Step 34: 211/267=7 remainder 24

Image may be NSFW.
Clik here to view.
Step 35: Place 7 on I.

Image may be NSFW.
Clik here to view.
Step 36: Place remainder 024 on JKL.

Image may be NSFW.
Clik here to view.
Step 37: 2437/267=9 remainder 34

Image may be NSFW.
Clik here to view.
Step 38: Place 9 on J.

Image may be NSFW.
Clik here to view.
Step 39: Place remainder 0034 on KLMN.

Image may be NSFW.
Clik here to view.
Step 40: Divide 627 by current root 89.

Image may be NSFW.
Clik here to view.
Step 41: 627/89=7 remainder 4. Place 7 on F as 3rd root.

Image may be NSFW.
Clik here to view.
Step 42: Place remainder 004 on GHI.

Image may be NSFW.
Clik here to view.
Step 43: Subtract 3rd root^2 from 49 on IJ. (-c^2)

Image may be NSFW.
Clik here to view.
Step 44: Place 49-7^2=00 on IJ.

Image may be NSFW.
Clik here to view.
Step 45: Subtract 3rd root^3 from 343 on MNO. (-c^3)

Image may be NSFW.
Clik here to view.
Step 46: Place 343-7^3=000 on MNO.

Image may be NSFW.
Clik here to view.
Step 47: Cube root of 721734273 is 897.

Image may be NSFW.
Clik here to view.
Final state: Answer 897

Abacus state transition. (Click to Zoom)
Image may be NSFW.
Clik here to view.
Image may be NSFW.
Clik here to view.


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


Please place your mouse on the buttons and click one by one. These are blog ranking sites.
Image may be NSFW.
Clik here to view.　Image may be NSFW.
Clik here to view.　Image may be NSFW.
Clik here to view.