Square Root of 610

√610 at a glance

Exact value

√610

Decimal (10 places)

24.6981780705

Rounded

24.7 · 24.70 · 24.698

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.698178

Prime factorization

2 × 5 × 61

Cube root

8.480926

How to simplify √610

The prime factorization of 610 is 2 × 5 × 61. Every prime appears only once, so there is no pair to bring outside the radical — √610 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 610, 2, 5 and 61 appear an odd number of times, so √610 is irrational and 24.6981780705 is a rounded value.

Where √610 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √610 lies between 24 and 25. 610 is 34 above 576 and 15 below 625, so the root is closer to 25.

• Straight line between 576 and 625: 24.6939 (0.02% low)
• Tangent from 24, i.e. 24 + 34 ÷ 48: 24.7083 (0.04% high)
• Tangent from 25, i.e. 25 − 15 ÷ 50: 24.7000 (0.01% high)

For √610 the tangent at 25 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 610 is just 15 below 625.

Finding √610 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √610 = 24.6981780705 to every decimal shown.

√610 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √610 the pattern is [24; 1, 2, 3, 5, 5, 3, 2, 1, 48] with the block of 9 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √610 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 − 610y² = 1. Its smallest solution in positive whole numbers is x = 10,323,982,819, y = 418,005,846. Because the period is odd, the equation with −1 on the right also has a solution: 71,847² − 610 × 2,909² = −1.

√610 in geometry and everyday measurements

• A square garage floor of 610 square feet measures about 24.7 ft (24 ft 8 in) per side, and its corner-to-corner diagonal is √1220 ≈ 34.9 ft.
• 610 = 9² + 23² = 13² + 21², so by the Pythagorean theorem √610 is the diagonal of rectangles measuring 9 × 23 and 13 × 21 — and the distance between the points (0, 0) and (9, 23) on a grid.

Square roots near √610 and related roots

• The cube root of 610 is about 8.480926.
• Squaring undoes the root: (√610)² = 610, while 610² = 372,100 — the number whose square root is 610.

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