Square Root of 629

√629 at a glance

Exact value

√629

Decimal (10 places)

25.0798724080

Rounded

25.1 · 25.08 · 25.080

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.079872

Prime factorization

17 × 37

Cube root

8.568081

How to simplify √629

The prime factorization of 629 is 17 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √629 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 629, 17 and 37 appear an odd number of times, so √629 is irrational and 25.0798724080 is a rounded value.

Where √629 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √629 lies between 25 and 26. 629 is 4 above 625 and 47 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.0784 (0.01% low)
• Tangent from 25, i.e. 25 + 4 ÷ 50: 25.0800 (0% high)
• Tangent from 26, i.e. 26 − 47 ÷ 52: 25.0962 (0.06% high)

For √629 the tangent at 25 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 629 is just 4 above 625.

Finding √629 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 629: following the tangent line down to zero simplifies to averaging x with 629 ÷ x.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √629 = 25.0798724080 to every decimal shown.

√629 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √629 the pattern is [25; 12, 1, 1, 12, 50] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √629 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 − 629y² = 1. Its smallest solution in positive whole numbers is x = 123,245,001, y = 4,914,100. Because the period is odd, the equation with −1 on the right also has a solution: 7,850² − 629 × 313² = −1.

√629 in geometry and everyday measurements

• A square garage floor of 629 square feet measures about 25.08 ft (25 ft 1 in) per side, and its corner-to-corner diagonal is √1258 ≈ 35.5 ft.
• 629 = 2² + 25² = 10² + 23², so by the Pythagorean theorem √629 is the diagonal of rectangles measuring 2 × 25 and 10 × 23 — and the distance between the points (0, 0) and (2, 25) on a grid.

Square roots near √629 and related roots

• The cube root of 629 is about 8.568081.
• Squaring undoes the root: (√629)² = 629, while 629² = 395,641 — the number whose square root is 629.

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