Cube roots

Odd numbers build cubes too, in runs:

Each run is as long as the number being cubed, and each starts where the last one stopped.

So the run 91 + 93 + 95 + 97 + 99 + 101 + 103 + 105 + 107 + 109 (ten odd numbers) must be 10³ = 1000, with no adding needed.

Since 8 = 2³, we call 2 the cube root of 8 and write 2 = ∛8. In general, if y = x³, then x is the cube root of y. So ∛27 = 3, ∛1000 = 10, and ∛(n³) = n.

Unlike a square root, a cube root has only one value: a negative number cubed stays negative, so no other number gives the same cube. (∛−1331 = −11.)

How can you tell if a number is a cube? Prime factorisation again, but into three identical groups instead of two: each triplet inside the cube-root sign comes out as one.

• 3375 = 3 × 3 × 3 × 5 × 5 × 5 = (3 × 5) × (3 × 5) × (3 × 5) = 15³. So yes, ∛3375 = 15. (Or as triplets: (3 × 3 × 3) × (5 × 5 × 5) = 3³ × 5³.)
• 500 = 2 × 2 × 5 × 5 × 5. The 5s make a triplet, but there are only two 2s. No three identical groups, so 500 is not a perfect cube.

The reason: each prime factor of a number appears three times over in the factorisation of its cube. 12 = 2 × 2 × 3, and 12³ = 1728 = 2³ × 2³ × 3³.

A handy check for a cube's last digit: 0, 1, 4, 5, 6 and 9 give cubes ending in the same digit (4³ = 64, 9³ = 729), while 2 and 8 swap (2³ = 8, 8³ = 512) and 3 and 7 swap (3³ = 27, 7³ = 343). So 4913 lies between 10³ and 20³ and ends in 3, which means its cube root ends in 7: it's 17.

A pattern to try: take the squares 1, 4, 9, 16, 25, 36 and write the differences, then the differences of those. After two levels they're all 2. Do the same with the cubes 1, 8, 27, 64, 125, 216 and see how many levels it takes.