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# Cube roots

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Simple procedures for approximating the cube root of a number

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### Cube roots

1. 1. Cube root estimates David Coulson, 2013 dtcoulson@gmail.com
2. 2. To estimate the cube root of a number in the range 100-500.... 250
3. 3. 250 Reduce it by a tenth... 225 To estimate the cube root of a number in the range 100-500....
4. 4. 250 Reduce it by a tenth... 225 Divide it by 100... 2.25 To estimate the cube root of a number in the range 100-500....
5. 5. 250 Reduce it by a tenth... 225 Divide it by 100... 2.25 ...and add four. 6.25 To estimate the cube root of a number in the range 100-500....
6. 6. 6.25 Correct value = 6.30 error = 0.8%
7. 7. Second example: 430 Reduce it by a tenth... 387 Divide it by 100... 3.87 Add four... 7.25 7.54 (4% error) ~
8. 8. 300 Reduce it by a tenth... 270 Divide it by 100... 2.70 Add four... 6.70 6.69 (0.1% error) Third example: ~
9. 9. 100 Reduce it by a tenth... 90 Divide it by 100... 0.90 Add four... 4.90 4.64 (6% error) Fourth example: ~
10. 10. Percenterror The technique works best at values close to 216.
11. 11. The technique comes from applying a straight line approximation to the cube curve in regions close to multiples of 3. x=3k
12. 12. Substitute x=3k
13. 13. Substitute
14. 14. Substitute
15. 15. Substitute
16. 16. In the case where k=1, the straight line is tangent at (3,27), and we have a reasonable approximation for the cube roots of numbers near 27.
17. 17. This version produces some very good estimates for cube roots in the range 10~50.
18. 18. (2.46, error 3%) This version produces some very good estimates for cube roots in the range 10~50. Dividing by 27 can be approximated as dividing by 30 and making the answer 10% bigger.
19. 19. (error 0.1%) Alternatively, when the numerator and denominator are of similar size, you can adjust the values by the same amount to get numbers that are easier to process.
20. 20. However, one number in three is on the 3 times table. Therefore it’s often easy to divide by 27. (2.76, error 1%)
21. 21. The case where k=2 shifts the tangent line up to (6, 216). The formula benefits from numbers which are multiples of three because the derivative term 3x2 resonates with the x3 term for these numbers.
22. 22. This works well for numbers near 216, extending out as far as 100~500. Approximating the denominator as 100 will often be good enough, but better results are obtained by reducing the numerator by ten percent first.
23. 23. What about other multiples of 3? Consider k=3. This should be good in the region near 729. Experimentation shows it is reliable between 500~1000. K values above 3 produce tangent points greater than 1000. Cube roots for numbers above 1000 can be related to numbers in the range 1~10. Therefore k values bigger than 3 are redundant.
24. 24. [END]