What is a cube?
The cube of a number is that number multiplied by itself three times. We write it with a small raised 3. For example, 4³ = 4 × 4 × 4 = 64. We read 4³ as "4 cubed". A whole number obtained by cubing another whole number is called a perfect cube. The first few perfect cubes are 1, 8, 27, 64, 125 and 216.
Key idea / Key idea
n³ = n × n × n. Cubing gives the volume of a cube with edge n. A cube of edge 5 cm has volume 5³ = 125 cm³.
What is a cube root?
The cube root of a number is the value that, when cubed, gives that number. The symbol is ∛. Since 3³ = 27, we say ∛27 = 3. Cube root is the inverse of cubing. A neat way to find a cube root is prime factorisation: group equal prime factors in threes and take one from each group.
Worked example / Worked example
Find ∛1728 by prime factorisation.
1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
Group in threes: (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3)
Take one from each group: 2 × 2 × 3 = 12.
Check: 12³ = 12 × 12 × 12 = 1728. ✔ So ∛1728 = 12.
Unlike square roots, a cube root can be negative. Because (−2)³ = −8, we have ∛(−8) = −2. Decimals work too: (0.2)³ = 0.008.
Remember / Remember
- Cubing and cube root are inverse operations.
- The cube of a negative number is negative.
- A cube uses three equal factors; a square uses two.
- Read ∛ as "cube root of".