Perfect Cubes and Cube Roots

What is a perfect cube?

A perfect cube is a whole number multiplied by itself three times. 5 × 5 × 5 = 125, so 125 is a perfect cube. We write it 5³ and read it five cubed.

In 5³, the 5 is the base and the 3 is the exponent. The exponent counts copies of the base: two copies for a square, three copies for a cube.

A cube with side 5 is built from 125 unit cubes. That 125 is the volume. The cube root reverses the question: given the volume, what's the side?

$\sqrt[3]{125} = 5$ because a cube of volume 125 has side 5.

Drag to change the side length. The same cube unfolds into six square faces: six because a cube has six sides, square because every face of a cube is a square.

Cubes grow fast. 5³ = 125, but 6³ = 216 and 10³ = 1 000. Five values (1³ through 5³) are worth memorising; past that, work it out.

Worksheet

Try these to put the idea into practice. The goal is recall for 1³ through 5³, and keeping cubes and squares straight in your head.

The curriculum expects fluent recall of 1³ through 5³. Drill them until they're automatic.

Going further

Cube roots of non-perfect cubes, like

$\sqrt[3]{50}$, sit between two whole numbers. $\sqrt[3]{50}$ is between 3 and 4, closer to 4. Estimating those values is the next step.

Cubes are one case of a more general shape. A rectangular box with length l, width w, and height h has volume l × w × h. When all three are equal, the box is a cube and the volume is s³.