Square Root of 596

√596 at a glance

Exact value

2√149

Decimal (10 places)

24.4131112315

Rounded

24.4 · 24.41 · 24.413

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.413111

Prime factorization

2² × 149

Cube root

8.415542

How to simplify √596

Look for the largest perfect square that divides 596. Here it is 4 (2²), because 596 = 4 × 149 and 149 has no square factor left:

The prime factorization tells the same story: 596 = 2² × 149. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 149 stays inside.

Check: (2√149)² = 2² × 149 = 4 × 149 = 596. As a decimal, 2√149 = 2 × 12.2065556157 ≈ 24.4131112315.

Where √596 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √596 lies between 24 and 25. 596 is 20 above 576 and 29 below 625, so the root is closer to 24.

• Straight line between 576 and 625: 24.4082 (0.02% low)
• Tangent from 24, i.e. 24 + 20 ÷ 48: 24.4167 (0.01% high)
• Tangent from 25, i.e. 25 − 29 ÷ 50: 24.4200 (0.03% high)

For √596 the tangent at 24 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 596 is just 20 above 576.

Finding √596 with the Babylonian method

Picture a rectangle with an area of 596 and one side x; the other side must be 596 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √596.

Start from the nearest whole number, 24 (24² = 576):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √596 = 24.4131112315 to every decimal shown.

√596 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √596 the pattern is [24; 2, 2, 2, 1, 1, 1, 6, 2, 1, 9, 12, 9, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √596 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 − 596y² = 1. Its smallest solution in positive whole numbers is x = 25,801,741,449, y = 1,056,880,510.

√596 in geometry and everyday measurements

• A square garage floor of 596 square feet measures about 24.41 ft (24 ft 5 in) per side, and its corner-to-corner diagonal is √1192 ≈ 34.5 ft.
• 596 = 14² + 20², so by the Pythagorean theorem √596 is the diagonal of a 14 × 20 rectangle — and the distance between the points (0, 0) and (14, 20) on a grid.
• Since √596 = 2√149, a length of √596 is exactly 2 copies of the length √149 laid end to end.

Square roots near √596 and related roots

• The cube root of 596 is about 8.415542.
• Because 596 = 4 × 149, the root is twice √149: 2 × 12.206556 ≈ 24.413111.

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