Square number

Because n dots along each side fill an n by n square, these are called square numbers, or perfect squares. So 1, 4, 9, 16, 25 are square numbers, since they are 1², 2², 3², 4², 5².

In real life: A square floor tiled with the same square tiles always needs a square number of tiles, for example a 4 by 4 area needs 16 tiles.

The odd number pattern

Each new odd number adds an L shaped layer around the square, turning one square into the next. So 1 = 1², 1 + 3 = 4 = 2², 1 + 3 + 5 = 9 = 3², and 1 + 3 + 5 + 7 = 16 = 4².

In real life: This pattern lets you build the next square from the one before by adding just one odd number, without multiplying again.

Worked example 1

Question. Show, using the odd number pattern, that 5² = 25.

Answer: So 5² = 25, matching the pattern that the sum of the first n odd numbers is n².

Square root

It undoes squaring. Since 4² = 16, we have √16 = 4. A perfect square has a whole number square root; other numbers fall between two whole roots. We take the positive root because a side length is never negative.

In real life: If a square garden has an area of 144 square metres, its side is √144 = 12 metres, so the square root turns an area back into a length.

Worked example 2

Question. Find √144.

Answer: √144 = 12.

Cube number

Because n small cubes along each edge fill an n by n by n cube, these are perfect cubes. So 1, 8, 27, 64, 125 are cube numbers, being 1³, 2³, 3³, 4³, 5³.

In real life: A storage box that is 3 units along every edge holds 3³ = 27 unit boxes, so cube numbers count how many small cubes fill a big cube.

Cube root

Since 3³ = 27, the cube root ∛27 = 3. Unlike square roots, a cube root of a positive number is a single positive value, and cube roots of negative numbers are allowed too.

In real life: If a cubic tank holds 343 cubic units, each edge is ∛343 = 7 units, so the cube root turns a volume back into an edge length.

Worked example 3

Question. Find ∛343.

Answer: ∛343 = 7.