Squares, cubes and roots Revision Notes for Edexcel GCSE Maths

Squares, cubes and roots

Squares and square roots

Square numbers are created when a whole number is multiplied by itself. These numbers have a special connection to geometry - they represent the areas of squares with whole number side lengths.

Understanding squares

When you multiply a number by itself, you get a square number. You can write this using index notation, where the small 2 shows you're squaring the number.

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Worked Example: Calculating Square Numbers

$2 \times 2 = 2^2 = 4$
$5 \times 5 = 5^2 = 25$
$9 \times 9 = 9^2 = 81$
$13 \times 13 = 13^2 = 169$

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The connection to area is helpful to remember: a square with sides of 5 cm has an area of $25 \text{ cm}^2$ , which is why 25 is a square number. This geometric relationship makes square numbers easier to visualise and understand.

Square roots

Square roots work in the opposite direction to squares. The square root symbol $\sqrt{}$ asks the question: "Which number, when squared, gives this result?"

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Worked Example: Finding Square Roots

$\sqrt{4} = 2$ (because $2^2 = 4$ )
$\sqrt{25} = 5$ (because $5^2 = 25$ )
$\sqrt{81} = 9$ (because $9^2 = 81$ )

chatImportant

You should memorise the square numbers up to $15 \times 15$ and their corresponding square roots, as these appear frequently in exams. This will save you valuable time and help you spot patterns in more complex problems.

Cubes and cube roots

Cube numbers are formed when a whole number is multiplied by itself, then multiplied by itself again. These numbers represent the volumes of cubes with whole number side lengths.

Understanding cubes

When you multiply a number by itself twice more, you get a cube number. You write this using index notation with a small 3.

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Worked Example: Calculating Cube Numbers

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

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Think of volume: a cube with edges of 3 cm has a volume of $27 \text{ cm}^3$ , which is why 27 is a cube number. This three-dimensional connection helps distinguish cubes from squares.

Cube roots

Cube roots are the opposite of cubes. The cube root symbol $\sqrt[3]{}$ asks: "Which number, when cubed, gives this result?"

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Worked Example: Finding Cube Roots

$\sqrt[3]{8} = 2$ (because $2^3 = 8$ )
$\sqrt[3]{27} = 3$ (because $3^3 = 27$ )
$\sqrt[3]{1000} = 10$ (because $10^3 = 1000$ )

chatImportant

You should memorise the cubes of 2, 3, 4, 5 and 10, along with their corresponding cube roots. These values appear regularly in exam questions and knowing them instantly will boost your confidence.

Key exam skills

Explaining mathematical statements

When exam questions ask you to explain whether a mathematical statement is correct, structure your answer clearly:

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Worked Example: Structured Mathematical Explanation

Question: "Is the statement 'A square number cannot be a prime number' correct?"

Step 1: State whether the statement is true or false Step 2: Give a clear reason starting with "Yes because..." or "No because..." Step 3: Support your reasoning with examples or mathematical facts

Answer: Yes, this statement is mostly correct because:

Prime numbers have exactly two factors (1 and themselves)
Square numbers have at least three factors (1, the number itself, and its square root)
The only exception is that 1 is not considered prime

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Always structure your explanations clearly with:

A definitive answer (true or false)
A clear reason starting with "Yes because..." or "No because..."
Supporting examples or mathematical facts

Working without a calculator

Practice calculating squares, cubes and roots without a calculator by memorising the key values. This builds confidence and saves time in exams.

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Common Exam Mistake: Students often lose marks by not showing their working clearly when calculating roots and powers. Always write out your steps, even for calculations you can do mentally.

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Key Points to Remember:

Square numbers come from multiplying a number by itself ( $n^2$ )
Cube numbers come from multiplying a number by itself twice more ( $n^3$ )
Square roots ( $\sqrt{}$ ) and cube roots ( $\sqrt[3]{}$ ) are the inverse operations
Square numbers relate to areas of squares, cube numbers relate to volumes of cubes
Memorise squares up to $15^2$ and cubes of 2, 3, 4, 5, and 10 for quick recall in exams

Squares, cubes and roots (Edexcel GCSE Maths): Revision Notes