Cube Root Calculator
Calculate the cube root (∛x) of any number, cube a number (x³), find the nth root, or check whether a number is a perfect cube. Includes simplified surd form. Full GCSE and A-Level maths guide with perfect cubes reference table.
Perfect Cubes Reference Table (1–30)
What Is a Cube Root?
The cube root of a number x is the value y such that y × y × y = x. In other words, y3 = x. We write the cube root using the radical symbol with a small 3: ∛x, or equivalently as x1/3.
Cube roots are related to cubing in the same way that square roots are related to squaring: they are inverse operations. If you cube a number and then take the cube root, you get back to where you started.
Cube Roots of Negative Numbers
A key difference from square roots: cube roots of negative numbers are real and negative. This is because when you multiply three negative numbers together, the result is negative: (−2) × (−2) × (−2) = −8. Therefore ∛(−8) = −2.
Compare with square roots: (−2) × (−2) = +4, so √(−4) has no real solution (it requires complex numbers). Cube roots do not have this restriction because odd powers preserve the sign of the base.
Perfect Cubes: The Numbers to Know
A perfect cube is a number obtained by cubing an integer. Memorising the first ten or twenty perfect cubes is enormously helpful in GCSE maths, as it allows you to quickly recognise and simplify cube roots mentally.
| Integer (n) | n³ | ∛(n³) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 8 | 2 |
| 3 | 27 | 3 |
| 4 | 64 | 4 |
| 5 | 125 | 5 |
| 6 | 216 | 6 |
| 7 | 343 | 7 |
| 8 | 512 | 8 |
| 9 | 729 | 9 |
| 10 | 1,000 | 10 |
| 15 | 3,375 | 15 |
| 20 | 8,000 | 20 |
| 25 | 15,625 | 25 |
How to Estimate Cube Roots Without a Calculator
To estimate ∛N without a calculator, find the two perfect cubes that N falls between, then interpolate. For example, to estimate ∛50:
- ∛27 = 3 and ∛64 = 4, so ∛50 is between 3 and 4.
- 50 is closer to 64 than to 27 (distance 14 vs 23 from the perfect cubes), so the answer is closer to 4.
- ∛50 ≈ 3.7 (actual value: 3.684...).
This technique is valuable in non-calculator GCSE exams where you need to give an estimate.
Simplifying Cube Roots (Surd Form)
A cube root is in simplified surd form when no perfect cube factor remains inside the radical. To simplify ∛N:
- Find the prime factorisation of N.
- Group factors in threes. Every group of three identical factors can come outside the radical as a single factor.
- Write the simplified form: k∛m where k is the number of complete groups.
Example: Simplify ∛54
54 = 2 × 27 = 2 × 33
∛54 = ∛(33 × 2) = 3∛2
So ∛54 = 3∛2 ≈ 3.78
Cube Root on a Calculator
On a scientific calculator, use the x1/y or ∛ button. Enter the number, press the root button, then enter 3. Some calculators have a dedicated cube root button. In spreadsheet software:
- Excel / Google Sheets:
=A1^(1/3)or=POWER(A1, 1/3) - Python:
x ** (1/3)(note: use-(-x)**(1/3)for negatives) orimport math; math.cbrt(x)(Python 3.11+) - JavaScript:
Math.cbrt(x)
Cube Roots in Geometry: Volume Problems
The most common real-world application of cube roots is finding the side length of a cube from its volume. If a cube has volume V, its side length is ∛V.
Example: Find the side of a cube with volume 729 cm³
Side = ∛729 = ∛(93) = 9 cm
Engineers and architects use cube roots when designing cubic containers, water tanks, and buildings with cubic proportions. In physics, cube roots appear in calculations involving volume-related quantities such as the radius of a sphere from its volume: r = ∛(3V / 4π).
Nth Roots
The nth root of x is the number y such that yn = x. It is written as n√x or x1/n. Common examples:
| Root | Symbol | Example |
|---|---|---|
| Square root | √x = x1/2 | √16 = 4 |
| Cube root | ∛x = x1/3 | ∛27 = 3 |
| Fourth root | 4√x = x1/4 | 4√81 = 3 |
| Fifth root | 5√x = x1/5 | 5√32 = 2 |
Even roots (square, fourth, sixth...) of negative numbers are not real. Odd roots (cube, fifth, seventh...) of negative numbers are real and negative.
Cube Roots in GCSE and A-Level Maths
At GCSE, cube roots appear in the topic of Powers and Roots. Key skills:
- Recognising and evaluating perfect cubes (1 to at least 103)
- Expressing cube roots as fractional indices (∛x = x1/3)
- Solving equations of the form x3 = k
- Estimating cube roots using knowledge of perfect cubes
At A-Level, cube roots appear in solving cubic equations, complex number theory, and calculus applications (e.g., optimisation problems involving volume).
Related Calculators
How the Cube Root Calculator Works
This calculator uses the current UK grading system and educational standards to help students, parents, and teachers understand academic performance. UK education follows specific grading frameworks that differ between GCSEs, A-Levels, and university degrees.
Understanding how grades are calculated and what they mean for future progression is important for making informed decisions about subject choices, university applications, and career planning.
Key Information for 2025/26
GCSEs in England use the 9-1 grading scale, where 9 is the highest and 4 is a standard pass (equivalent to the old C grade). A-Levels use the A*-E scale with UCAS tariff points ranging from 56 (A*) to 16 (E). Universities typically require grades between AAA and CCC depending on the course and institution, with highly competitive courses often asking for A*A*A.
Example Calculation
A student achieving grades of A*, A, and B at A-Level would earn UCAS tariff points of 56 + 48 + 40 = 144 points total. This exceeds the typical entry requirement for most Russell Group universities (128 points or AAB equivalent) and would make the student competitive for many courses.
Source: Based on Ofqual and UCAS 2025/26 guidelines. Last updated March 2026.