What is the cube root of 72?

The cube root of 72 is approximately 4.1602. Since 72 is not a perfect cube, the result is an irrational number.

How do you calculate the cube root of 72?

You can compute the cube root using Newton's Method, an iterative algorithm. Starting from an initial estimate, refine it via x = (2·x + a/x²) / 3 until the value stabilizes. Convergence is quadratic — each iteration roughly doubles the digits of precision.

Is ∛72 rational or irrational?

∛72 is irrational. Its decimal expansion (4.1602...) never terminates or repeats.

Cube Root of 72 = 4.1602

Quick Answer

∛72 = 4.1602 (irrational)

Because 4.1602³ ≈ 72

Cube Root Calculator

Cube Root Simplify Cube Root Cubed (n³) Is N a Perfect Cube? Exponents (bⁿ)

Use the calculator above to find the cube root of any real number (positive, negative, or decimal). See below the step-by-step method using Newton's iteration, plus a table of common perfect cubes.

Step-by-Step: Cube Root of 72

The cube root of 72 can be approximated using Newton's Method with the iterative formula:

x_n+1 = (2 · x_n + a / x_n²) / 3

Starting from x₀ = 24.0000 (rough estimate for 72):

Iteration	x	x³
1	16.041667	4128.0834
2	10.787708	1255.4157
3	7.398036	404.9014
4	5.370533	154.9002

After 4 iterations the value converges to ∛72 ≈ 4.1602.

Why does this converge?

Newton's Method is a root-finding algorithm. For ∛a, we solve x³ − a = 0. The iteration uses the derivative (3x²) to refine each estimate. Convergence is quadratic — each iteration roughly doubles the digits of precision.

Formula

The cube root of a number a is the value x such that:

x³ = a ⇔ ∛a = x

Every real number has exactly one real cube root. Unlike the square root, the cube root is defined for negative inputs: ∛(−a) = −∛a.

How to Calculate Cube Roots

For perfect cubes (8, 27, 64, 125, …) the cube root is an integer — just recall the cube of small integers.

For other numbers, use Newton's Method:

Start with a rough estimate (e.g. x₀ = a / 3).
Apply x_n+1 = (2x_n + a/x_n²) / 3.
Repeat 3–4 times — the value converges fast (each step roughly doubles the digits of precision).

Nearby Cube Roots

∛n	Result	Type
∛20	2.7144	irrational
∛24	2.8845	irrational
∛25	2.9240	irrational
∛27	3.0000	perfect cube
∛30	3.1072	irrational
∛32	3.1748	irrational
∛40	3.4200	irrational
∛48	3.6342	irrational
∛50	3.6840	irrational
∛64	4.0000	perfect cube
∛75	4.2172	irrational
∛100	4.6416	irrational

Perfect Cube Roots Table

Common perfect cubes from 1 to 1000. For the full table 1–100, see our cube root chart 1 to 1000.

n	∛n
1	1
8	2
27	3
64	4
125	5
216	6
343	7
512	8
729	9
1000	10

Want the Simplified Radical Form?

The decimal value above is useful for measurements, but in algebra problems you often need the simplified radical form. Try our Simplify Cube Root Calculator to see ∛72 in simplest form (e.g. 2∛3 instead of 4.1602).