A square number fills a flat square. A cube number fills a solid cube. Eight small blocks stack into a 2 by 2 by 2 cube, and twenty seven make a 3 by 3 by 3 one.
Cubing and cube rooting
| Operation | What it does | Example |
|---|---|---|
| Cubing | Multiply a number by itself three times | 4³ = 4 × 4 × 4 = 64 |
| Cube rooting | Find the number that was cubed | cube root of 64 is 4 |
4 cubed is 4 times 4 times 4, which is 64. It is not 4 times 3, and it is not 4 plus 4 plus 4. The small 3 counts how many copies are being multiplied, not added.
The cube numbers
The first ten cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000. They grow far faster than the squares: by the time you reach 10, the square is 100 but the cube is already 1000.
Double the side of a square and the area becomes four times bigger. Double the side of a cube and the volume becomes eight times bigger. Lengths scale by k, areas by k squared, volumes by k cubed.
A useful difference from squares
Cubing 0 to 9 gives last digits 0, 1, 8, 7, 4, 5, 6, 3, 2, 9. Every digit appears exactly once, so unlike squares, the last digit never rules a cube out. Instead it tells you exactly what the last digit of the root must be.
Split a six digit number into blocks of three from the right: 262 and 144. The 144 ends in 4, and only 8 cubed ends in 4, so the root ends in 8. The 262 sits between 6 cubed and 7 cubed, so the root begins with 6. The answer is 68.
Cube roots by prime factorisation
The method is the same as for square roots with one change: group the prime factors into threes instead of pairs, and take one factor from each group of three.
Minus 2 cubed is minus 8, because three negative signs multiplied leave a negative. So every number has exactly one cube root, and the cube root of a negative number is negative.
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