Cubes
Calculate cubes like 12³ or 23³ in seconds.
Core Concepts
Finding the cube of a 2-digit number (ab) uses the expansion of (a + b)³.
Formula: a³ + 3a²b + 3ab² + b³.
The Ratio Method
Write four terms in a row:
a³ | a²b | ab² | b³
Then double the middle two terms and add them up!
Example: 12³
a=1, b=2.
Row 1: 1 | 2 | 4 | 8
Row 2: - | 4 | 8 | - (Double middle)
Add: 1 | 7 | 2 | 8
Solved Examples
Example 1: 12³
Question: Calculate 12³.
• a=1, b=2.
• Terms: 1³, 1²×2, 1×2², 2³ → 1, 2, 4, 8.
• Double middles: 4, 8.
• Add: 1 | (2+4) | (4+8) | 8.
• 1 | 6 | 12 | 8.
• Carry over the 1 from 12: 1 | (6+1) | 2 | 8.
• Answer: 1728.
Example 2: 21³
Question: Calculate 21³.
• a=2, b=1.
• Terms: 8, 4, 2, 1.
• Double middles: 8, 4.
• Add: 8 | (4+8) | (2+4) | 1.
• 8 | 12 | 6 | 1.
• Carry over: (8+1) | 2 | 6 | 1.
• Answer: 9261.
Example 3: 11³
Question: Calculate 11³.
• a=1, b=1.
• Terms: 1, 1, 1, 1.
• Double middles: 2, 2.
• Add: 1 | 3 | 3 | 1.
• Answer: 1331.
