Square Root of 592

√592 at a glance

Exact value

4√37

Decimal (10 places)

24.3310501212

Rounded

24.3 · 24.33 · 24.331

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.331050

Prime factorization

2⁴ × 37

Cube root

8.396673

How to simplify √592

Look for the largest perfect square that divides 592. Here it is 16 (4²), because 592 = 16 × 37 and 37 has no square factor left:

The prime factorization tells the same story: 592 = 2⁴ × 37. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 37 stays inside.

592 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √592 = 2√148, and √148 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√37)² = 4² × 37 = 16 × 37 = 592. As a decimal, 4√37 = 4 × 6.0827625303 ≈ 24.3310501212.

Where √592 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √592 lies between 24 and 25. 592 is 16 above 576 and 33 below 625, so the root is closer to 24.

• Straight line between 576 and 625: 24.3265 (0.02% low)
• Tangent from 24, i.e. 24 + 16 ÷ 48: 24.3333 (0.01% high)
• Tangent from 25, i.e. 25 − 33 ÷ 50: 24.3400 (0.04% high)

For √592 the tangent at 24 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 592 is just 16 above 576.

Finding √592 with the Babylonian method

Picture a rectangle with an area of 592 and one side x; the other side must be 592 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √592.

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 √592 = 24.3310501212 to every decimal shown.

√592 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √592 the pattern is [24; 3, 48] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √592 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 − 592y² = 1. Its smallest solution in positive whole numbers is x = 73, y = 3.

√592 in geometry and everyday measurements

• A square garage floor of 592 square feet measures about 24.33 ft (24 ft 4 in) per side, and its corner-to-corner diagonal is √1184 ≈ 34.4 ft.
• 592 = 4² + 24², so by the Pythagorean theorem √592 is the diagonal of a 4 × 24 rectangle — and the distance between the points (0, 0) and (4, 24) on a grid.
• Since √592 = 4√37, a length of √592 is exactly 4 copies of the length √37 laid end to end.

Square roots near √592 and related roots

• The cube root of 592 is about 8.396673.
• Because 592 = 4 × 148, the root is twice √148: 2 × 12.165525 ≈ 24.33105.

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