What is the cube root of 210?

The cube root of 210 is approximately 5.9439. Since 210 is not a perfect cube, the result is an irrational number.

How do you calculate the cube root of 210?

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 ∛210 rational or irrational?

∛210 is irrational. Its decimal expansion (5.9439...) never terminates or repeats.

Cube Root of 210 = 5.9439

Quick Answer

∛210 = 5.9439 (irrational)

Because 5.9439³ ≈ 210

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 210

The cube root of 210 can be approximated using Newton's Method with the iterative formula:

x_n+1 = (2 · x_n + a / x_n²) / 3

Starting from x₀ = 70.0000 (rough estimate for 210):

Iteration	x	x³
1	46.680952	101722.9915
2	31.152758	30233.5753
3	20.840634	9051.7543
4	14.054923	2776.4214

After 4 iterations the value converges to ∛210 ≈ 5.9439.

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
∛48	3.6342	irrational
∛50	3.6840	irrational
∛64	4.0000	perfect cube
∛75	4.2172	irrational
∛100	4.6416	irrational
∛125	5.0000	perfect cube
∛128	5.0397	irrational
∛150	5.3133	irrational
∛200	5.8480	irrational
∛216	6.0000	perfect cube
∛250	6.2996	irrational
∛343	7.0000	perfect cube

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 ∛210 in simplest form (e.g. 2∛3 instead of 5.9439).