The cube root

The idea

If y = x³, then x is the cube root of y, written x = ∛y; in general ∛n³ = n.

A box of 27 cubes

A cube-shaped box is packed exactly with 27 unit cubes. There is no space left over.

How many unit cubes long is the edge of the box?

• 3
• 9
• 13

Build it, read the edge

Use − and + to build a cube that uses exactly the number of unit cubes asked for. The side you find is the cube root.

Back from the cube

If y = x³, then x is the cube root of y, written x = ∛y. In general, ∛n³ = n.

Cubing and taking the cube root undo each other: cube 3 to get 27, and ∛27 takes you back to 3. The cube root is the edge of the cube made from y unit cubes.

• Cube y = x³ | Cube root x = ∛y
• 8 = 2³ | ∛8 = 2
• 27 = 3³ | ∛27 = 3
• 64 = 4³ | ∛64 = 4
• 1000 = 10³ | ∛1000 = 10

Notes

If y = x³, then x is the cube root of y, written x = ∛y. In general, ∛n³ = n.

Check yourself

Which is true?

Find ∛125.

Answer: 5

5 × 5 × 5 = 125, so ∛125 = 5.

Find ∛(9^3).

Answer: 9

∛(n³) = n, so ∛(9³) = 9.

A cube-shaped box holds exactly 1000 unit cubes. How long is its edge?

Answer: 10 units

10 × 10 × 10 = 1000, so the edge is ∛1000 = 10 units.

• ∛64 = 8 because 8 × 8 = 64. That uses two factors, so 8 is the square root. For ∛ we need three: 4 × 4 × 4 = 64.
• ∛64 = 4 because 4 × 4 × 4 = 64 — correct. Yes! 4 × 4 × 4 = 64, so ∛64 = 4.
• ∛64 = 21 because 64 ÷ 3 ≈ 21. Dividing by 3 does not undo a cube. 21 × 21 × 21 is far more than 64.