Square Root of 634

√634 at a glance

Exact value

√634

Decimal (10 places)

25.1793566240

Rounded

25.2 · 25.18 · 25.179

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.179357

Prime factorization

2 × 317

Cube root

8.590724

How to simplify √634

The prime factorization of 634 is 2 × 317. Every prime appears only once, so there is no pair to bring outside the radical — √634 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 634, 2 and 317 appear an odd number of times, so √634 is irrational and 25.1793566240 is a rounded value.

Where √634 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √634 lies between 25 and 26. 634 is 9 above 625 and 42 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.1765 (0.01% low)
• Tangent from 25, i.e. 25 + 9 ÷ 50: 25.1800 (0% high)
• Tangent from 26, i.e. 26 − 42 ÷ 52: 25.1923 (0.05% high)

For √634 the tangent at 25 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 634 is just 9 above 625.

Finding √634 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 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √634 = 25.1793566240 to every decimal shown.

√634 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √634 the pattern is [25; 5, 1, 1, 2, 1, 4, 3, 6, 1, 7, 1, 1, …] with the block of 23 terms after the semicolon repeating forever (only the first 12 of the 23 are shown). A pattern that never ends is one more proof that √634 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 − 634y² = 1. Its smallest solution in positive whole numbers is x = 8,711,856,945,587,257,031,251, y = 345,992,039,259,400,361,250 — 22 digits for x, even though 634 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 65,999,458,125² − 634 × 2,621,173,333² = −1.

√634 in geometry and everyday measurements

• A square garage floor of 634 square feet measures about 25.18 ft (25 ft 2 in) per side, and its corner-to-corner diagonal is √1268 ≈ 35.6 ft.
• 634 = 3² + 25², so by the Pythagorean theorem √634 is the diagonal of a 3 × 25 rectangle — and the distance between the points (0, 0) and (3, 25) on a grid.

Square roots near √634 and related roots

• The cube root of 634 is about 8.590724.
• Squaring undoes the root: (√634)² = 634, while 634² = 401,956 — the number whose square root is 634.

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