Squares, Square Roots, and Cubes Revision Notes for Grade 10 NSC Matric Mathematical Literacy

Squares, Square Roots, and Cubes

What are squares?

When we square a number, we multiply that number by itself. This operation is fundamental in mathematics and has many practical applications.

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Definition: A square is the result when a number is multiplied by itself.

The mathematical notation for squaring uses a small 2 as a superpower: $n^2$ (read as "n squared").

Visual representation of squares

We can understand squares better by looking at them as grid diagrams. The number of blocks along each side of the square represents the number being squared.

Looking at these grids:

A $2 \times 2$ grid contains $2^2 = 4$ small squares
A $4 \times 4$ grid contains $4^2 = 16$ small squares
A $5 \times 5$ grid contains $5^2 = 25$ small squares
A $10 \times 10$ grid contains $10^2 = 100$ small squares

Connection to area

The area of any square equals the side length multiplied by itself. This is why we say "side squared" when calculating the area of a square.

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Formula: Area of square = side × side = $\text{side}^2$

This concept will be essential when working with area calculations in geometry.

What are square roots?

A square root is the opposite operation of squaring. When we find the square root, we ask: "What number, when multiplied by itself, gives us this result?"

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Definition: The square root of a number is the value that, when squared, produces the original number.

We write square roots using the symbol $\sqrt{}$ . For example: $\sqrt{16} = 4$ because $4^2 = 16$ .

Understanding square roots through squares

Finding the square root of a number is the same as finding the side length of a square with that area. If a square has an area of 16 small squares, then each side must be 4 units long, so $\sqrt{16} = 4$ .

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Key relationship: Squaring and taking square roots are inverse operations - they undo each other.

What are cubes?

A cube is formed when we multiply a number by itself three times. This creates a three-dimensional shape.

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Definition: To cube a number means to multiply it by itself twice (three factors total).

The mathematical notation uses a small 3 as a superpower: $n^3$ (read as "n cubed").

Visual representation of cubes

We can visualise cubes as three-dimensional blocks made up of smaller unit cubes.

Looking at these cube diagrams:

A $2 \times 2 \times 2$ cube contains $2^3 = 8$ small cubes
A $3 \times 3 \times 3$ cube contains $3^3 = 27$ small cubes
A $5 \times 5 \times 5$ cube contains $5^3 = 125$ small cubes

Key formulas

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To square a number: Multiply it by itself

Example: $2 \times 2 = 2^2 = 4$

To cube a number: Multiply it by itself twice

Example: $2 \times 2 \times 2 = 2^3 = 8$

Using your calculator

You can easily calculate square roots using your calculator. Simply enter the number and press the square root key ( $\sqrt{}$ button).

For squares and cubes, you can:

Use the power button (usually $x^y$ or $\wedge$ )
For squares: enter the number, then $x^2$ or use the $x^2$ button
For cubes: enter the number, then $x^3$

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Most scientific calculators have dedicated buttons for common operations like squares and square roots, making calculations much faster than doing them by hand.

Worked examples

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Worked Example: Finding squares

Calculate $6^2$ :

$6^2 = 6 \times 6 = 36$

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Worked Example: Finding square roots

Calculate $\sqrt{49}$ :

Ask: "What number multiplied by itself equals 49?"

$7 \times 7 = 49$ , so $\sqrt{49} = 7$

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Worked Example: Finding cubes

Calculate $4^3$ :

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

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Worked Example: Practical application

A square garden has an area of 144 square metres. What is the length of each side?

Solution: Find $\sqrt{144} = 12$ metres

The garden has sides of 12 metres each.

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Worked Example: Volume calculation

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

Solution: Volume = $3^3 = 3 \times 3 \times 3 = 27$ cubic centimetres

The box has a volume of 27 cm³.

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Key Points to Remember:

Squaring means multiplying a number by itself: $n^2 = n \times n$
Square roots are the inverse of squaring - they help us find the original number
Cubing means multiplying a number by itself three times: $n^3 = n \times n \times n$
Squares relate to area (2D), while cubes relate to volume (3D)
Use your calculator's square root button for quick calculations of $\sqrt{}$ values

Squares, Square Roots, and Cubes (Grade 10 NSC Matric Mathematical Literacy): Revision Notes