Squares, cubes and roots Revision Notes for AQA GCSE Maths

Squares, cubes and roots

Squares and square roots

Square numbers are created when a number is multiplied by itself. You can write square numbers using index notation, where the small ² shows that the number is squared.

For example, $5 \times 5 = 5^2 = 25$ . This means that 25 is a square number.

Square numbers have a practical meaning in the real world - they represent the areas of squares with whole number side lengths. A square with sides of 5 cm has an area of 25 cm².

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Real-world connection: Square numbers aren't just abstract mathematical concepts. Every time you calculate the area of a square room, garden, or any square shape, you're working with square numbers!

Square roots are the opposite operation of squaring. The square root symbol √ is used to show this. When you find the square root of a number, you're asking "what number multiplied by itself gives this result?"

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Worked Example: Finding Square Roots

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

chatImportant

You need to memorise the square numbers up to $15 \times 15 = 225$ and their corresponding square roots for your exam. This is essential knowledge that cannot be looked up!

Cubes and cube roots

Cube numbers are created when a number is multiplied by itself twice. You can write cube numbers using index notation with a small ³.

For example, $3 \times 3 \times 3 = 3^3 = 27$ . This means that 27 is a cube number.

Cube numbers represent the volumes of cubes with whole number side lengths. A cube with sides of 3 cm has a volume of 27 cm³.

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Visual understanding: Just like square numbers relate to flat areas, cube numbers relate to 3D volumes. Think of building blocks or dice - their volumes are cube numbers when their sides are whole numbers.

Cube roots are the opposite operation of cubing. The cube root symbol ∛ is used to show this. When you find the cube root of a number, you're asking "what number multiplied by itself twice gives this result?"

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Worked Example: Finding Cube Roots

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

chatImportant

You need to memorise the cubes of 2, 3, 4, 5, and 10, along with their corresponding cube roots. These appear frequently in exam questions.

Working with squares, cubes and roots

Understanding the connection to prime numbers

Square numbers can never be prime numbers (except for some special cases). This is because a square number must have a factor other than 1 and itself - namely, its square root. For instance, 9 has factors 1, 3, and 9, so it cannot be prime.

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Why squares aren't prime: Think about it logically - if $n^2$ is a square number, then $n$ is always a factor of $n^2$ (other than 1 and the number itself). This automatically disqualifies it from being prime!

Common exam questions

You should be able to calculate squares, cubes, and roots without using a calculator for the basic values you've memorised. Practice questions typically ask you to:

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Typical Exam Questions

Work out squares like $6^2 = 36$
Find cube roots like $\sqrt[3]{125} = 5$
Calculate cubes like $4^3 = 64$
Find square roots like $\sqrt{121} = 11$

Exam tips

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Essential Exam Strategy

Learn your square numbers up to $15^2 = 225$
Memorise cube numbers: $2^3 = 8$ , $3^3 = 27$ , $4^3 = 64$ , $5^3 = 125$ , $10^3 = 1000$
Remember that square roots and cube roots are the inverse operations
When explaining your reasoning, use phrases like "Yes because..." or "No because..." and give mathematical justification
Always check if your answer makes sense by working backwards

Remember!

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Key Points to Remember

Square numbers are formed by multiplying a number by itself ( $n \times n = n^2$ )
Cube numbers are formed by multiplying a number by itself twice ( $n \times n \times n = n^3$ )
Square roots (√) and cube roots (∛) are the opposite operations of squaring and cubing
Square numbers relate to areas of squares, cube numbers relate to volumes of cubes
You must memorise key square and cube numbers for your GCSE exam - practice them regularly!

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