How do you calculate the cube root?

The cube root of x is the number y such that y³ = x. On a scientific calculator, use the x^(1/3) button or the ∛ button. In Excel, use =x^(1/3) or =POWER(x,1/3). Mentally, you can estimate by finding the two perfect cubes that the number falls between. For example, ∛50 is between ∛27=3 and ∛64=4, so roughly 3.7.

Cube Root Calculator | ∛x & Perfect Cubes

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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 x^1/3.

∛x = x1/3 = y   where   y³ = x

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 x^1/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) or import 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 x^1/n. Common examples:

Root	Symbol	Example
Square root	√x = x^1/2	√16 = 4
Cube root	∛x = x^1/3	∛27 = 3
Fourth root	⁴√x = x^1/4	⁴√81 = 3
Fifth root	⁵√x = x^1/5	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 = x^1/3)
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).

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Frequently Asked Questions

The cube root of 27 is 3, because 3 × 3 × 3 = 27. Written in mathematical notation: ∛27 = 3 or 27^1/3 = 3. 27 is a perfect cube (3³).

The cube root of x is the number y such that y³ = x. On a scientific calculator, use the x^1/3 button or the ∛ button. In Excel, use =x^(1/3) or =POWER(x,1/3). To estimate mentally, find the two perfect cubes your number falls between, then interpolate. For example, ∛50 is between ∛27=3 and ∛64=4, so approximately 3.7.

A perfect cube is a number that is the cube (third power) of an integer. The first ten perfect cubes are: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. A number is a perfect cube if its cube root is exactly an integer.

Yes! Unlike square roots, cube roots of negative numbers are real. The cube root of a negative number is negative. For example, ∛(−8) = −2, because (−2) × (−2) × (−2) = −8. This is because an odd number of negative factors always gives a negative product.

The cube root of 125 is 5, because 5 × 5 × 5 = 125. Written as: ∛125 = 5 or 125^1/3 = 5. 125 is a perfect cube (5³).

To simplify ∛N, find the largest perfect cube that is a factor of N. Write N = k³ × m where k is as large as possible. Then ∛N = k∛m. For example, ∛54 = ∛(27 × 2) = 3∛2. If N is itself a perfect cube, the cube root simplifies to an integer.

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