Cubes & Cube Roots — Class 8 IMO

Perfect Cubes

A perfect cube is a number that is the cube of a whole number. The first few are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000. The cube of a number n is n × n × n.

Example 1: What is 4³?

4 × 4 × 4 = 64.

Example 2: What is 5³?

5 × 5 × 5 = 125.

Quick recap

A perfect cube = the cube of a whole number.
Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

✓ Quick check

What is 3³?

3 × 3 × 3 = 27.

Which of these is a perfect cube?

27 = 3³.

Finding Cubes by Prime Factorisation

To check whether a number is a perfect cube, write its prime factorisation and group the primes in threes. If every prime forms complete groups of three, the number is a perfect cube.

Example 1: What is 6³?

6 × 6 × 6 = 216.

Example 2: What is 10³?

10 × 10 × 10 = 1000.

Quick recap

Group prime factors in threes to test for a perfect cube.
n³ = n × n × n.

✓ Quick check

What is 7³?

7 × 7 × 7 = 343.

What is 8³?

8 × 8 × 8 = 512.

Cube Roots & Volume Problems

The cube root reverses cubing: the cube root of 125 is 5 because 5³ = 125. The volume of a cube of side a is a³, so a cube of side 4 has volume 64.

Example 1: What is the cube root of 125?

5, since 5³ = 125.

Example 2: A cube has side 4 cm. What is its volume?

4³ = 64 cm³.

Quick recap

Cube root reverses cubing (∛125 = 5).
Volume of a cube = side³.

✓ Quick check

What is the cube root of 216?

6³ = 216.

What is the cube root of 343?

7³ = 343.