What is the simplified form of ∛189?

The cube root of 189 simplifies to 3∛7. In decimal form, ∛189 ≈ 5.7388.

How do you simplify the cube root of 189?

To simplify ∛189, find the prime factorization of 189, group the factors in triples (groups of three), and move each complete triple outside the radical sign. Any factors not part of a complete triple remain inside.

Is ∛189 rational or irrational?

∛189 is irrational. Although it can be written in the simplified radical form 3∛7, its decimal expansion (5.7388...) never terminates or repeats.

Simplify the Cube Root of 189 = 3∛7

Quick Answer

∛189 = 3∛7 (≈ 5.7388)

Formula: 189 = 3³ × 7 → ∛189 = 3∛7

Simplify Cube Root Calculator

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

Use the calculator above to simplify any cube root to its simplest radical form (e.g. ∛24 = 2∛3). See below the step-by-step method using prime factorization, plus a table of common simplified cube roots.

Step-by-Step: Simplify ∛189

To simplify ∛189, find the largest perfect cube that divides 189:

189 = 27 × 7
∛189 = ∛27 × ∛7
∛189 = 3∛7

In decimal form: ∛189 ≈ 5.7388

See prime factorization details

Prime factorization: 189 = 3³ × 7

Group the identical primes in triples (three of a kind); each complete triple contributes one factor outside the radical, and any leftovers (1 or 2 of the same prime) stay inside.

Formula

To rewrite any cube root in simplest radical form:

∛(a³ · b) = a · ∛b

Where:

a³ = the largest perfect-cube factor of the radicand (the number under the ∛ symbol)
b = the remaining cubefree factor (no perfect cube divides it other than 1)

Worked example for ∛24:

24 = 8 × 3 = 2³ × 3
So a³ = 8, a = 2, b = 3
Therefore ∛24 = 2∛3

How to Simplify Cube Roots

Simplifying a cube root means rewriting it in the simplest radical form, where the number under the radical (the radicand) has no perfect cube factors other than 1.

Method: Prime Factorization

Find the prime factorization of the radicand.
Identify triples (groups of three) of identical prime factors.
Each complete triple contributes one factor outside the radical.
Any prime factor not part of a complete triple (1 or 2 leftovers) stays under the radical.

Example: Simplify ∛128

128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁷
We have two complete triples of 2's (2³ × 2³) plus a leftover 2.
Each triple → contributes one 2 outside. Two triples → 2 × 2 = 4 outside. The leftover 2 stays inside.
∛128 = 4∛2

Nearby Simplified Cube Roots

Click any value to see the full step-by-step solution.

∛n	Simplified	≈ Decimal
∛96	2∛12	4.5789
∛108	3∛4	4.7622
∛125	5	5.0000
∛128	4∛2	5.0397
∛135	3∛5	5.1299
∛162	3∛6	5.4514
∛189	3∛7	5.7388
∛192	4∛3	5.7690
∛216	6	6.0000
∛243	3∛9	6.2403
∛250	5∛2	6.2996
∛256	4∛4	6.3496

Want the Decimal Value Instead?

If you want the decimal approximation (e.g. ∛24 ≈ 2.8845) with the Newton's Method step-by-step, use our Cube Root Calculator.

Is 189 Itself a Perfect Cube?

If the radicand 189 is itself a perfect cube (e.g. 216 = 6³), then ∛189 simplifies to an integer. Check with our Perfect Cube Checker for the verdict + prime-factorization proof.

References

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