worldofcalcsCube Root Calculator
Cube root
∛1,728 = 12
1,728 = 2⁶ × 3³
1,728 is a perfect cube: 12 × 12 × 12 = 1,728.
What is a cube root?
The cube root of a number is the value that, multiplied by itself three times, gives the number: ∛27 = 3 because 3 × 3 × 3 = 27. Unlike square roots, negative numbers have real cube roots: ∛−8 = −2. This calculator gives the cube root, checks for perfect cubes using prime factors as in the Class 8 'Cubes and Cube Roots' chapter, and writes other roots in simplest form.
Prime factorisation method
∛(a³ × b³ × c) = a × b × ∛c
Write the number as a product of primes and group equal primes in threes. Each group of three comes out of the root as one factor; anything left over stays inside.
Good to know
- A perfect cube's prime factors all come in groups of three.
- The cube of an even number is even and of an odd number is odd.
- The last digit of a cube tells you the last digit of its cube root: cubes ending in 8 have roots ending in 2, and cubes ending in 7 have roots ending in 3.
- 1729 = 1³ + 12³ = 9³ + 10³ is famous as the Hardy–Ramanujan number, the smallest number that is the sum of two cubes in two ways.
- The cube root of a volume gives the side of a cube with that volume: a 1,000-litre tank shaped as a cube is 1 m on each side.
Worked example: ∛1728 and ∛54
- 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2⁶ × 3³
- Group in threes: (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3)
- ∛1728 = 2 × 2 × 3 = 12, so 1728 is a perfect cube
- 54 = 2 × 3³, so ∛54 = 3∛2 ≈ 3.7798
Cubes of 1 to 20
Perfect cubes worth knowing.
| n | n³ | n | n³ |
|---|---|---|---|
| 1 | 1 | 11 | 1,331 |
| 2 | 8 | 12 | 1,728 |
| 3 | 27 | 13 | 2,197 |
| 4 | 64 | 14 | 2,744 |
| 5 | 125 | 15 | 3,375 |
| 6 | 216 | 16 | 4,096 |
| 7 | 343 | 17 | 4,913 |
| 8 | 512 | 18 | 5,832 |
| 9 | 729 | 19 | 6,859 |
| 10 | 1,000 | 20 | 8,000 |
Read any n³ backwards to get its cube root.