Cubes
The cube of a number is the number multiplied by itself three times: n³ = n × n × n. The name comes from geometry: a cube with side n units has volume n³ cubic units.
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
n³ | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |
Numbers like 1, 8, 27 and 64, which are cubes of whole numbers, are called perfect cubes (or cube numbers). Learning the first ten makes cube-root work much faster.
Patterns in cubes
The cube of an even number is even; the cube of an odd number is odd.
The cube of a negative number is negative: (−3)³ = −27.
Units digits: a number ending in 2 has a cube ending in 8, and one ending in 8 has a cube ending in 2. Likewise, 3 ↔ 7. Numbers ending in 0, 1, 4, 5, 6 and 9 have cubes ending in the same digit.
Sums of consecutive odd numbers: 1 = 1³, 3 + 5 = 8 = 2³, 7 + 9 + 11 = 27 = 3³, 13 + 15 + 17 + 19 = 64 = 4³.
The cube of a fraction: (2/3)³ = 8/27.
Testing for a perfect cube
Write the number as a product of primes and group equal primes in threes. The number is a perfect cube only if every prime can be grouped into complete triplets.
Example. Is 1728 a perfect cube?
1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3).
All primes form triplets, so yes: 1728 = (2 × 2 × 3)³ = 12³.
Example. Is 500 a perfect cube?
500 = 2 × 2 × 5 × 5 × 5. The 5s make a triplet, but the two 2s do not. So 500 is not a perfect cube.
The smallest multiplier or divisor
To make a number a perfect cube:
multiply by the primes needed to complete each triplet, or
divide by the primes left over.
Example. 500 = 2 × 2 × 5³. To complete the triplet of 2s we need one more 2, so multiply by 2: 1000 = 10³. Or divide by 4 (= 2 × 2): 125 = 5³.
Example. 392 = 2 × 2 × 2 × 7 × 7. Multiply by 7 to get 2744 = 14³.
Cube roots
The cube root of a number x is the number whose cube is x. It is written ∛x.
∛27 = 3, since 3³ = 27. ∛(−64) = −4, since (−4)³ = −64.
By prime factorisation: take one prime from each triplet and multiply.
Example. ∛3375. 3375 = (3 × 3 × 3) × (5 × 5 × 5), so ∛3375 = 3 × 5 = 15.
A quick estimation method (for perfect cubes up to 6 digits)
Split the number into groups of three digits from the right. For 17576, the groups are 17 and 576.
The units digit of the root comes from the last group's units digit. 576 ends in 6, so the root ends in 6.
The tens digit comes from the other group: find the largest cube not exceeding 17. That is 8 = 2³, so the tens digit is 2.
So ∛17576 = 26. Check: 26³ = 17576. ✓
This trick works only when you already know the number is a perfect cube.
Summary
n³ = n × n × n; the first ten cubes are worth memorising.
A number is a perfect cube when its prime factors group completely into triplets.
The cube root takes one factor from each triplet; cube roots of negative numbers are negative.