A cube number is found by multiplying a whole number by itself, then by itself again: 2 × 2 × 2 = 8, so 8 is a perfect cube. The cube root is the inverse: ∛8 = 2. At KS3, recognising the first six perfect cubes saves real exam time and speeds up fraction and volume calculations.
What is a cube number?
A cube number is the result of multiplying an integer by itself, and then by itself once more. The operation is called cubing, and is written using a superscript 3: 4³ means 4 × 4 × 4 = 64.
The name comes directly from geometry. Build a cube whose edge length is 4 units and its volume is 4 × 4 × 4 = 64 cubic units. This is why volume is measured in cm³ or m³ — the superscript 3 records that three equal lengths have been multiplied together. A cube number always represents a genuine three-dimensional volume.
It is easy to confuse cubing with squaring. 4² = 4 × 4 = 16, whereas 4³ = 4 × 4 × 4 = 64. The key difference is the number of times you use 4 as a factor: twice for squaring, three times for cubing.
Which perfect cubes should you learn by heart?
Learning the first ten perfect cubes lets you answer cube-root questions without a calculator and spots patterns in harder number problems instantly.
| n | n³ |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1000 |
A useful self-check: the units digits of the cubes cycle through 1, 8, 7, 4, 5, 6, 3, 2, 9, 0 and then repeat. If you cube a number ending in 4, your answer must end in 4; if it ends in anything else, you have made an arithmetic slip.
What is a cube root and how do you write it?
The cube root of a number asks: which value, cubed, gives this result? It uses the radical symbol with a small 3 in the crook: ∛.
∛27 = 3, because 3³ = 27. ∛125 = 5, because 5³ = 125.
Worked example — step by step:
Find ∛512.
- Locate 512 in the perfect-cube table: 8³ = 512.
- Therefore ∛512 = 8.
- Verify: 8 × 8 = 64; 64 × 8 = 512. ✓
On a scientific calculator the cube root function is often labelled ∛x or accessed via SHIFT and the x³ key. Always verify the result by cubing it — a quick mental or calculator check takes three seconds and catches most mistakes.
Can you cube a negative number?
Yes. Cubing a negative number always gives a negative result because multiplying three negative numbers together produces a negative:
(−3)³ = (−3) × (−3) × (−3) = 9 × (−3) = −27
This is an important contrast with squaring: (−3)² = +9, because two negative signs cancel. Cube roots of negative numbers also exist and are negative: ∛(−27) = −3.
Watch the notation in exam questions carefully. −3³ means −(3³) = −27, not (−3)³, because the convention is to apply the power before the negative sign unless brackets force otherwise.
How do you estimate the cube root of a non-perfect cube?
When the number is not a perfect cube, bound it between two known cubes and close in by trial.
Worked example: estimate ∛200 to one decimal place.
- Identify the surrounding perfect cubes: 5³ = 125, 6³ = 216. So 5 < ∛200 < 6.
- 200 is much closer to 216 than to 125, so try values closer to 6.
- Try 5.8: 5.8 × 5.8 = 33.64; 33.64 × 5.8 = 195.1 — too small.
- Try 5.9: 5.9 × 5.9 = 34.81; 34.81 × 5.9 = 205.4 — too large.
- Since 200 lies between 195.1 and 205.4, ∛200 ≈ 5.8 (to 1 d.p.).
This bounding technique mirrors the trial and improvement method and is all that is expected at KS3. A calculator gives ∛200 ≈ 5.848, confirming our estimate rounds correctly.
How do cube numbers appear in maths problems?
Cube numbers and cube roots come up in several KS3 and GCSE contexts beyond simply being asked to recall them:
- Volume of a cube: if a cube has volume 343 cm³, its edge length is ∛343 = 7 cm.
- Index notation: 10³ = 1000, 10⁶ = 1 000 000 — recognising powers of 10 as cubes links to standard form and place value.
- Prime factor decomposition: 8 = 2³ and 27 = 3³. Writing a composite number with a cube in its prime factorisation (e.g. 216 = 2³ × 3³) allows you to simplify cube roots of fractions, such as ∛(8/27) = 2/3.
- Algebra: expressions like (2x)³ = 8x³ and the difference of two cubes a³ − b³ appear at GCSE Higher.
Frequently asked questions
Is 0 a cube number?
Yes. 0³ = 0 × 0 × 0 = 0, so zero is a perfect cube and ∛0 = 0. Most KS3 exam questions focus on positive integers, but zero satisfies the definition perfectly. It is the only number that is simultaneously a square number, a cube number, and its own square root and cube root.
What is the difference between squaring and cubing?
Squaring multiplies a number by itself once (n × n = n²), producing results that grow moderately: 2² = 4, 3² = 9, 4² = 16. Cubing multiplies by itself twice more (n × n × n = n³), growing considerably faster: 2³ = 8, 3³ = 27, 4³ = 64. Critically, cubing preserves the sign of negative numbers whereas squaring always gives a non-negative result.
Does every number have a cube root?
Every real number has exactly one real cube root, including negative numbers and zero. This differs from square roots, which only exist for non-negative numbers in the real number system. So ∛(−8) = −2 is perfectly valid, whereas √(−8) is not a real number.
How do cube numbers differ from square numbers?
Square numbers are found by multiplying a number by itself once: 1, 4, 9, 16, 25, 36 … Cube numbers are found by multiplying a number by itself twice: 1, 8, 27, 64, 125, 216 … Some numbers are both — 1 and 64 appear in both lists (1 = 1² = 1³; 64 = 8² = 4³). These overlaps sometimes feature in problem-solving questions at the end of KS3 papers.
For guided KS3 number practice with Professor Pi, visit aitutors.me.