What is the simplified form of √6174?

The square root of 6174 simplifies to 21√14. In decimal form, √6174 ≈ 78.5748.

How do you simplify the square root of 6174?

To simplify √6174, find the prime factorization of 6174, group the factors in pairs, and move each pair outside the radical sign. Any unpaired factors remain inside.

Is √6174 rational or irrational?

√6174 is irrational. Although it can be written in the simplified radical form 21√14, its decimal expansion (78.5748...) never terminates or repeats.

Simplify the Square Root of 6174 = 21√14

Quick Answer

√6174 = 21√14 (≈ 78.5748)

Formula: 6174 = 2 × 3² × 7³ → √6174 = 21√14

Simplify Square Root Calculator

Simplify √ Decimal √ Squared (n²) Is N a Perfect Square?

Use the calculator above to simplify any square root to its simplest radical form (e.g. √50 = 5√2). See below the step-by-step method using prime factorization, plus a table of common simplified square roots.

Step-by-Step: Simplify √6174

To simplify √6174, find the largest perfect square that divides 6174:

6174 = 441 × 14
√6174 = √441 × √14
√6174 = 21√14

In decimal form: √6174 ≈ 78.5748

See prime factorization details

Prime factorization: 6174 = 2 × 3² × 7³

Pair the identical primes; each pair contributes one factor outside the radical, and any leftover primes stay inside.

Formula

To rewrite any square root in simplest radical form:

√(a² · b) = a · √b

Where:

a² = the largest perfect-square factor of the radicand (the number under the √ symbol)
b = the remaining squarefree factor (no perfect square divides it other than 1)

Worked example for √72:

72 = 36 × 2 = 6² × 2
So a² = 36, a = 6, b = 2
Therefore √72 = 6√2

How to Simplify Square Roots

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

Method: Prime Factorization

Find the prime factorization of the radicand.
Identify pairs of identical prime factors.
Each pair contributes one factor outside the radical.
Any unpaired prime factor stays under the radical.

Example: Simplify √72

72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
We have a pair of 2's, a pair of 3's, and a leftover 2.
Pair of 2's → contributes one 2 outside. Pair of 3's → contributes one 3 outside. Leftover 2 stays inside.
√72 = (2 × 3)√2 = 6√2

Nearby Simplified Square Roots

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

√n	Simplified	≈ Decimal
√343	7√7	18.5203
√360	6√10	18.9737
√392	14√2	19.7990
√405	9√5	20.1246
√450	15√2	21.2132
√500	10√5	22.3607
√512	16√2	22.6274
√625	25	25.0000
√720	12√5	26.8328
√800	20√2	28.2843
√900	30	30.0000
√1000	10√10	31.6228

Want the Decimal Value Instead?

If you want the decimal approximation (e.g. √50 ≈ 7.0711) with the Babylonian Method step-by-step, use our Square Root Calculator.

Is 6174 Itself a Perfect Square?

If the radicand 6174 is itself a perfect square (e.g. 144 = 12²), then √6174 simplifies to an integer. Check with our Perfect Square Checker for the verdict + prime-factorization proof.

References

Popular Simplified Square Roots

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