Cubes and n³

The idea

A number obtained by multiplying a number by itself three times is a cube, written n × n × n = n³; 1, 8, 27, … are perfect cubes, and fractions, decimals and negative numbers also have cubes.

A big cube of little cubes

A big cube is built from small unit cubes. Its edge is 4 unit cubes long.

How many unit cubes make the big cube?

• 12
• 16
• 64

Build and spread

Use − and + to change the side. Press Spread layers to pull the cube into layers. How many layers? How many cubes in each?

Three times over

A number multiplied by itself three times is a cube. We write n × n × n = n³, read “n cubed”.

The cube with edge 4 has 4 layers. Each layer is a square of 4 × 4 = 16 unit cubes. So the total is 4 × 4 × 4 = 64.

Is 9 a cube? 2 × 2 × 2 = 8 and 3 × 3 × 3 = 27, so 9 is not. Nor is any number from 10 to 26.

Fractions, decimals and negative numbers have cubes too: (4/6)^3 = 64/216, (13.08)^3 = 2237.810112 and (−6)^3 = (−6) × (−6) × (−6) = −216.

• n | n × n × n | n³
• 1 | 1 × 1 × 1 | 1
• 2 | 2 × 2 × 2 | 8
• 3 | 3 × 3 × 3 | 27
• 4 | 4 × 4 × 4 | 64
• 5 | 5 × 5 × 5 | 125

Notes

A number obtained by multiplying a number by itself three times is a cube: n × n × n = n³. 1, 8, 27, 64, 125, … are perfect cubes. Fractions, decimals and negative numbers can be cubed too.

Check yourself

Which of these is a perfect cube?

A cube has an edge of 6 unit cubes. How many unit cubes make it?

Answer: 216 cubes

One layer is 6 × 6 = 36 cubes and there are 6 layers: 36 × 6 = 216.

Find (2/3)^3.

Answer: 8/27

Top: 2 × 2 × 2 = 8. Bottom: 3 × 3 × 3 = 27. So (2/3)^3 = 8/27.

Find (−3)^3.

Answer: -27

(−3) × (−3) = 9, then 9 × (−3) = −27.

• 9. 9 = 3 × 3 uses 3 only twice. 2 × 2 × 2 = 8 and 3 × 3 × 3 = 27, so 9 is not a cube.
• 12. 12 is not one number multiplied by itself three times. 2 × 2 × 2 = 8 and 3 × 3 × 3 = 27.
• 27 — correct. Yes! 3 × 3 × 3 = 27.
• 36. 36 = 6 × 6 is a square. 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so 36 is not a cube.