Technique explanation
For a perfect cube, group digits in threes from the right. The last digit of the cube uniquely determines the last digit of the root. The leftmost group determines the first digit. For a three-digit root, use magnitude to bound the middle digit and verify with digit sums.
This method is only for perfect cubes. Standard fallback: Prime factorization or iterative estimation for non-perfect cubes.
Worked examples
Example 1: Cube root of 1860867
- Group: 1 | 860 | 867. Three-digit root.
- Last Digit: Cube ends in 7, so root ends in 3.
- First Digit: Left group is 1. 1^3 = 1, so first digit is 1.
- Middle Digit: Between 120^3 (1.72M) and 130^3 (2.19M), so middle digit is 2.
- Result: 123.
- Check: 123 × 123 × 123 = 1,860,867.
Example 2: Cube root of 13312053
- Group: 13 | 312 | 053. Three-digit root.
- Last Digit: Cube ends in 3, so root ends in 7.
- First Digit: Left group is 13. 2^3=8, so first digit is 2.
- Middle Digit: Between 230^3 (12.1M) and 240^3 (13.8M), so middle digit is 3.
- Result: 237.
- Check: 237 × 237 × 237 = 13,312,053.
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