Cubes & Cube Roots (Prime Factorisation) | Mathematics

Chapter 1 · Number Operations

1.1.2 Cubes & Cube Roots

A cube is a number multiplied by itself three times. Finding a cube root means breaking a number down into its prime factors and looking for matching groups of three.

🎯 By the end of this you can…

✓Cube any whole number by direct multiplication
✓Find a cube root using prime factorisation
✓Find the smallest number a value must be multiplied by to become a perfect cube

💡 The idea

Cube = multiply by itself three times

$n^{3} = n \times n \times n$ . A cube root asks the reverse question: what number, cubed, gives this? For a square root you paired digits in twos; for a cube root you'll group prime factors in threes instead.

✏️ Worked example

Find 28³

1
$2 8^{3} = 28 \times 28 \times 28$
2
$28 \times 28 = 784$ , then $784 \times 28 = 21952$

✅ Answer: $2 8^{3} = 21952$

📋 Cube root by prime factorisation

1
Divide the number repeatedly by the smallest prime that works (2, then 3, then 5, 7, 11…) until you reach 1.
2
Group the resulting prime factors into sets of three IDENTICAL primes.
3
Each complete group of three becomes one copy of that prime in the answer. Multiply those together.

🌳 ∛42875 by prime factorisation — one division at a time

⭐ 5 XP

42875 \div 5 = 8575

💡Why group in THREES, not twos? A cube root undoes

n \times n \times n

— three copies multiplied together — so only a complete matching triple of the same prime can be pulled back out as one factor.

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✏️ Worked example

Find ∛12167

1
$12167 \div 23 = 529$ , $529 \div 23 = 23$ , $23 \div 23 = 1$
2
Three identical 23s: $12167 = 23 \times 23 \times 23 = 2 3^{3}$

✅ Answer: $312167 = 23$

⚠️ The classic trap

Grouping prime factors in twos is a square-root habit that sneaks into cube-root questions. A cube root needs groups of three matching primes — two of the same prime left over is not enough to pull one out of the root.

🧮 Smallest number to multiply 4608 by, to make it a perfect cube

Prime-factorise 4608 by repeated division by 2, then 3:

4608 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^{9} \times 3^{2}

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🎮 Cube it, or cube-root it

⭐ 0 XP🔥 streak 0

Find

3 3^{3}

Find

4 5^{3}

Find

374088

Find

1 5^{3}

Find

39261

Smallest number to multiply 88 by, to make it a perfect cube?

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🤔 74088 = 2³ × 3³ × 7³. What is ∛74088?

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You finished this lesson!

Prime factorisation now unlocks both square and cube roots — the only thing that changes between them is whether you're hunting for pairs or for triples.

Keep reading — this lesson marks itself done automatically.