Cube Properties
A cube preserves a unique last digit, which is what makes it so useful in CAT. The unit digit of n³ depends only on the unit digit of n, and the map is: 0→0, 1→1, 4→4, 5→5, 6→6, 9→9 stay put, while 2↔8 and 3↔7 swap. So a cube can end in any digit 0–9 (unlike a square, which never ends in 2, 3, 7 or 8). To find a cube root by hand, group the digits in threes from the right. The number of groups gives the digit-count of the root; the last group fixes the unit digit of the root (read the swap map backwards), and the leading group fixes the tens digit by finding the largest cube not exceeding it. The other key property: in prime factorisation, a number is a perfect cube only if every exponent is a multiple of 3. That single test answers "is it a cube?" and "what is the smallest multiplier to make it a cube?" in one stroke.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.
Formula Reference Sheet
Cubes & cube roots
| Cube of n | n³ = n × n × n |
|---|---|
| Cube root | ∛(n³) = n |
| Perfect-cube test (prime factors) | every prime exponent is a multiple of 3 |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) |
| Sum of cubes | a³ + b³ = (a + b)(a² − ab + b²) |
Sum-of-cubes identities
| Sum of first n cubes | 1³ + 2³ + … + n³ = [n(n+1)/2]² |
|---|---|
| Link to triangular number | 1³ + 2³ + … + n³ = (1 + 2 + … + n)² |
| Sum of first n cubes | = [Σn]² where Σn = n(n+1)/2 |
| Cube of a binomial | (a + b)³ = a³ + 3a²b + 3ab² + b³ |
| Identity (a+b+c=0) | a³ + b³ + c³ = 3abc |