Squares and Cubes of Numbers Revision Notes for Grade 12 NSC Matric Mathematical Literacy

Squares and Cubes of Numbers

What are squares of numbers?

Squaring a number means multiplying that number by itself. When we write $4^2$ , we read this as "four squared" and it means $4 \times 4$ .

infoNote

The square of a number is the result you get when you multiply the number by itself. This fundamental operation forms the basis for understanding area calculations and many other mathematical concepts.

For example:

$2^2 = 2 \times 2 = 4$
$3^2 = 3 \times 3 = 9$
$5^2 = 5 \times 5 = 25$

Visual representation of squares

We can understand squares better by looking at them as grids. The number being squared tells us how many blocks to place along each side of a square grid.

infoNote

Visual representations help us understand why we call this operation "squaring" - because the result can be arranged in a perfect square shape!

For $2^2$ , we create a $2 \times 2$ grid with 4 small squares total.

For $4^2$ , we create a $4 \times 4$ grid with 16 small squares total.

The diagram shows that $5^2$ creates a $5 \times 5$ grid with 25 squares, and $10^2$ creates a $10 \times 10$ grid with 100 squares.

Key examples of perfect squares

lightbulbExample

Perfect Squares Demonstration:

$1^2 = 1 \times 1 = 1$
$2^2 = 2 \times 2 = 4$
$3^2 = 3 \times 3 = 9$
$4^2 = 4 \times 4 = 16$
$5^2 = 5 \times 5 = 25$
$10^2 = 10 \times 10 = 100$

chatImportant

Exam tip: You need to know how to square numbers to work with area calculations. Memorise the perfect squares from $1^2$ to $12^2$ for quick calculations.

What are square roots?

The square root is the opposite operation of squaring. If $4^2 = 16$ , then the square root of 16 is 4. We write this as $\sqrt{16} = 4$ .

chatImportant

Finding the square root of a number means finding which number, when multiplied by itself, gives you the original number. This is the inverse operation of squaring.

Key square roots to remember

$\sqrt{4} = 2$ (because $2^2 = 4$ )
$\sqrt{9} = 3$ (because $3^2 = 9$ )
$\sqrt{16} = 4$ (because $4^2 = 16$ )
$\sqrt{25} = 5$ (because $5^2 = 25$ )
$\sqrt{100} = 10$ (because $10^2 = 100$ )

Using a calculator for square roots

To find square roots on your calculator, enter the number and then press the square root key ( $\sqrt{}$ ). This is particularly useful for numbers that are not perfect squares.

infoNote

Exam tip: Finding the square root of a number is the same as finding the side length of a square when you know its area.

What are cubes of numbers?

Cubing a number means multiplying that number by itself three times. When we write $3^3$ , we read this as "three cubed" and it means $3 \times 3 \times 3$ .

infoNote

The cube of a number is the result you get when you multiply the number by itself three times. This operation is fundamental for understanding volume calculations.

Visual representation of cubes

Just like squares create flat grids, cubes create three-dimensional shapes. The number being cubed tells us how many small cubes to place along each edge.

This diagram shows three different cubes:

$2^3 = 2 \times 2 \times 2 = 8$ (a $2 \times 2 \times 2$ cube)
$3^3 = 3 \times 3 \times 3 = 27$ (a $3 \times 3 \times 3$ cube)
$5^3 = 5 \times 5 \times 5 = 125$ (a $5 \times 5 \times 5$ cube)

Key examples of perfect cubes

lightbulbExample

Perfect Cubes Demonstration:

$1^3 = 1 \times 1 \times 1 = 1$
$2^3 = 2 \times 2 \times 2 = 8$
$3^3 = 3 \times 3 \times 3 = 27$
$4^3 = 4 \times 4 \times 4 = 64$
$5^3 = 5 \times 5 \times 5 = 125$
$10^3 = 10 \times 10 \times 10 = 1000$

chatImportant

Exam tip: Cubes are used when working with volume calculations. The cube tells you how many small unit cubes fit inside the larger cube.

Worked examples

lightbulbExample

Example 1: Calculating squares

Calculate $8^2$ .

Solution: $8^2 = 8 \times 8 = 64$

lightbulbExample

Example 2: Finding square roots

Find $\sqrt{49}$ .

Solution: We need to find which number multiplied by itself gives 49. Since $7 \times 7 = 49$ , we know that $\sqrt{49} = 7$

lightbulbExample

Example 3: Calculating cubes

Calculate $4^3$ .

Solution: $4^3 = 4 \times 4 \times 4$ $4^3 = 16 \times 4 = 64$

lightbulbExample

Example 4: Using squares for area

A square garden has sides of 6 metres. What is its area?

Solution: Area = side × side = side² Area = $6^2 = 6 \times 6 = 36$ square metres

lightbulbExample

Example 5: Using cubes for volume

A cube-shaped box has edges of 3 cm. What is its volume?

Solution: Volume = length × width × height = edge³ Volume = $3^3 = 3 \times 3 \times 3 = 27$ cubic centimetres

bookmarkSummary

Key Points to Remember:

Squaring means multiplying a number by itself twice: $n^2 = n \times n$
Cubing means multiplying a number by itself three times: $n^3 = n \times n \times n$
Square roots are the opposite of squares: if $a^2 = b$ , then $\sqrt{b} = a$
Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) and perfect cubes (1, 8, 27, 64, 125) are important to memorise
Squares relate to area calculations, while cubes relate to volume calculations

Squares and Cubes of Numbers (Grade 12 NSC Matric Mathematical Literacy): Revision Notes