Square Root of 626

√626 at a glance

Exact value

√626

Decimal (10 places)

25.0199920064

Rounded

25.0 · 25.02 · 25.020

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.019992

Prime factorization

2 × 313

Cube root

8.554437

How to simplify √626

The prime factorization of 626 is 2 × 313. Every prime appears only once, so there is no pair to bring outside the radical — √626 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 626, 2 and 313 appear an odd number of times, so √626 is irrational and 25.0199920064 is a rounded value.

Where √626 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √626 lies between 25 and 26. 626 is 1 above 625 and 50 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.0196 (0% low)
• Tangent from 25, i.e. 25 + 1 ÷ 50: 25.0200 (0% high)
• Tangent from 26, i.e. 26 − 50 ÷ 52: 25.0385 (0.07% high)

For √626 the tangent at 25 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 626 is just 1 above 625.

Finding √626 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):

Because the starting guess was already close, two steps are enough to match √626 = 25.0199920064 to every decimal shown.

√626 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √626 the pattern is [25; 50] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 626 is one more than a perfect square (25² + 1). A pattern that never ends is one more proof that √626 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 − 626y² = 1. Its smallest solution in positive whole numbers is x = 1,251, y = 50. Because the period is odd, the equation with −1 on the right also has a solution: 25² − 626 × 1² = −1.

√626 in geometry and everyday measurements

• A square garage floor of 626 square feet measures about 25.02 ft (25 ft) per side, and its corner-to-corner diagonal is √1252 ≈ 35.4 ft.
• 626 = 1² + 25², so by the Pythagorean theorem √626 is the diagonal of a 1 × 25 rectangle — and the distance between the points (0, 0) and (1, 25) on a grid.

Square roots near √626 and related roots

• The cube root of 626 is about 8.554437.
• Squaring undoes the root: (√626)² = 626, while 626² = 391,876 — the number whose square root is 626.

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