Technique explanation
For a two-digit number ab (where a is tens and b is units), the cube is calculated using a geometric progression with ratio b/a: a^3 | a^2b | ab^2 | b^3. Write these four terms, double the two middle terms, add them, and resolve carries from right to left.
This method is limited to two-digit numbers for mental ease but can be extended. Standard fallback: Conventional long multiplication or the binomial expansion (10a + b)³.
Worked examples
Example 1: 23 cubed
- Identify: a=2, b=3. Ratio b/a = 1.5.
- GP Row: 8, 12, 18, 27.
- Double Middle: 24, 36.
- Sum: 8 | 36 | 54 | 27.
- Carries: 27 -> 7 (carry 2); 54+2=56 -> 6 (carry 5); 36+5=41 -> 1 (carry 4); 8+4=12.
- Result: 12167.
- Check: 23 × 23 × 23 = 12,167.
Example 2: 34 cubed
- Identify: a=3, b=4. Ratio 4/3.
- GP Row: 27, 36, 48, 64.
- Double Middle: 72, 96.
- Sum: 27 | 108 | 144 | 64.
- Carries: 64 -> 4 (carry 6); 144+6=150 -> 0 (carry 15); 108+15=123 -> 3 (carry 12); 27+12=39.
- Result: 39304.
- Check: 34 × 34 × 34 = 39,304.
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