Finding cubes and cube roots with prime factors and identities
A perfect cube is a number obtained by multiplying an integer by itself three times, such as 27 = 3³ and 1728 = 12³. To test whether a number is a perfect cube, write its prime factorisation and group the factors in threes. For 3375 = 3³ × 5³, every prime forms a complete triple, so its cube root is 3 × 5 = 15. If a factor is left over, the number is not a perfect cube, and the leftover tells you the smallest number to multiply or divide by to make it one.
Two identities speed up work with cubes: a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²). Also, (a + b)³ = a³ + b³ + 3ab(a + b), and whenever a + b + c = 0, a³ + b³ + c³ = 3abc. Unit digits help as well: the cube of a number ending in 2 ends in 8, and the cube of a number ending in 8 ends in 2. For a number that is not a perfect cube, the root lies between two consecutive integers, so ∛30 is a little more than 3 because 3³ = 27 and 4³ = 64.
Why it matters
Cubes appear whenever volume is involved, because a cube of side a has volume a³, and finding the side from a known volume needs a cube root. Fast factorisation with the cube identities saves time in Class 9 and 10 algebra and in competitive exams, and the same identities are used to simplify expressions in calculus and coordinate geometry.
Key terms
- Perfect cube
- A whole number that equals some integer multiplied by itself three times.
- Cube root
- The number which, when cubed, gives the original number.
- Prime factorisation
- Writing a number as a product of prime numbers only.