Cube Root Table

What Is a Cube Root?

The cube root of x is the number y such that y³ = x. It is written as ∛x or x^(1/3). Unlike square roots, cube roots are defined for negative numbers, since the cube of a negative number is negative.

• Notation: ∛x = y means y³ = x.
• Negative values: ∛(−x) = −∛x. For example, ∛(−27) = −3.
• Perfect cubes: integers whose cube roots are also integers, e.g., 1, 8, 27, 64, 125.
• Geometric meaning: ∛V is the side length of a cube with volume V.

Cube Root Table (1 to 100)

Decimal cube roots from 1 to 100, with the exact integer cube root shown for perfect cubes (highlighted rows).

Perfect Cubes Reference (1³ to 20³)

The first 20 perfect cubes. Memorizing these makes mental estimation of cube roots much easier.

How to Estimate Cube Roots Without a Calculator

To approximate ∛n manually, bracket n between two consecutive perfect cubes and refine.

1. Find the closest perfect cubes. For example, to estimate ∛50, note that 27 (3³) and 64 (4³) bracket 50, so ∛50 is between 3 and 4.
2. Use linear interpolation. ∛n ≈ a + (n − a³) / (3a²), where a is the cube root of the lower perfect cube.
3. Example for ∛50: with a = 3 and a³ = 27, ∛50 ≈ 3 + (50 − 27) / (3 × 9) = 3 + 23/27 ≈ 3.852. The actual value is 3.6840, so the approximation is close enough for estimation.

Properties of Cube Roots

• Defined for all real numbers, including negatives — unlike square roots, which require non-negative inputs.
• Multiplicative: ∛(a · b) = ∛a · ∛b.
• Quotient: ∛(a / b) = ∛a / ∛b for b ≠ 0.
• Power rule: ∛(a^n) = a^(n/3).
• Inverse of cubing: (∛x)³ = x and ∛(x³) = x.
• Irrationality: ∛n is irrational unless n is a perfect cube.

Applications of Cube Roots

Geometry

• Volume to side length: a cube of volume V has side length ∛V.
• Sphere radius from volume: r = ∛(3V / 4π).
• Scaling laws: when volume scales by a factor k, linear dimensions scale by ∛k.

Physics and Engineering

• Density and mass-volume relationships often involve cube roots when solving for length scales.
• Heat dissipation and other surface-to-volume problems use cube root scaling.
• Allometric scaling in biology relates organism size to metabolic rate through cube root laws.

Statistics and Data

• Cube-root transformations are used to normalize skewed data, preserving sign for negative values.

Frequently Asked Questions

What is the cube root of a negative number?

Cube roots of negative numbers are real and negative. For example, ∛(−64) = −4, because (−4)³ = −64. This is different from square roots, where negative inputs require imaginary numbers.

Is ∛2 rational?

No. ∛2 is irrational — its decimal expansion is non-terminating and non-repeating (∛2 ≈ 1.2599210). In general, ∛n is irrational unless n is a perfect cube.

How can I quickly check if a number is a perfect cube?

Look at the last digit of n. The last digit of n³ matches a unique digit for n: 0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9. Then estimate n by bracketing with the cube root table.

What is the difference between a cube and a cube root?

The cube of n is n³ (n × n × n). The cube root of n is the number whose cube equals n. They are inverse operations: ∛(n³) = n and (∛n)³ = n.

References

• Khan Academy — Introduction to cube roots.
• Wikipedia — Cube root (definition, properties, and history).
• OEIS A000578 — The sequence of cubes (perfect cubes).

See also