Square Root of 604

√604 at a glance

Exact value

2√151

Decimal (10 places)

24.5764114549

Rounded

24.6 · 24.58 · 24.576

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.576411

Prime factorization

2² × 151

Cube root

8.453028

How to simplify √604

Look for the largest perfect square that divides 604. Here it is 4 (2²), because 604 = 4 × 151 and 151 has no square factor left:

The prime factorization tells the same story: 604 = 2² × 151. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 151 stays inside.

Check: (2√151)² = 2² × 151 = 4 × 151 = 604. As a decimal, 2√151 = 2 × 12.2882057274 ≈ 24.5764114549.

Where √604 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √604 lies between 24 and 25. 604 is 28 above 576 and 21 below 625, so the root is closer to 25.

• Straight line between 576 and 625: 24.5714 (0.02% low)
• Tangent from 24, i.e. 24 + 28 ÷ 48: 24.5833 (0.03% high)
• Tangent from 25, i.e. 25 − 21 ÷ 50: 24.5800 (0.01% high)

For √604 the tangent at 25 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 604 is just 21 below 625.

Finding √604 with the Babylonian method

Picture a rectangle with an area of 604 and one side x; the other side must be 604 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √604.

Start from the nearest whole number, 25 (25² = 625):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √604 = 24.5764114549 to every decimal shown.

√604 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √604 the pattern is [24; 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 1, 4, …] with the block of 44 terms after the semicolon repeating forever (only the first 12 of the 44 are shown). A pattern that never ends is one more proof that √604 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 604y² = 1. Its smallest solution in positive whole numbers is x = 5,972,991,296,311,683,199, y = 243,037,569,063,951,720 — 19 digits for x, even though 604 is small, which is what makes Pell’s equation famous.

√604 in geometry and everyday measurements

• A square garage floor of 604 square feet measures about 24.58 ft (24 ft 7 in) per side, and its corner-to-corner diagonal is √1208 ≈ 34.8 ft.
• 604 is not a sum of two whole-number squares — the prime factor 151 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √604 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √604 as its space diagonal.
• Since √604 = 2√151, a length of √604 is exactly 2 copies of the length √151 laid end to end.

Square roots near √604 and related roots

• The cube root of 604 is about 8.453028.
• Because 604 = 4 × 151, the root is twice √151: 2 × 12.288206 ≈ 24.576411.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.