Square Root of 638

√638 at a glance

Exact value

√638

Decimal (10 places)

25.2586618806

Rounded

25.3 · 25.26 · 25.259

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.258662

Prime factorization

2 × 11 × 29

Cube root

8.608753

How to simplify √638

The prime factorization of 638 is 2 × 11 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √638 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 638, 2, 11 and 29 appear an odd number of times, so √638 is irrational and 25.2586618806 is a rounded value.

Where √638 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √638 lies between 25 and 26. 638 is 13 above 625 and 38 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.2549 (0.01% low)
• Tangent from 25, i.e. 25 + 13 ÷ 50: 25.2600 (0.01% high)
• Tangent from 26, i.e. 26 − 38 ÷ 52: 25.2692 (0.04% high)

For √638 the tangent at 25 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 638 is just 13 above 625.

Finding √638 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 √638 = 25.2586618806 to every decimal shown.

√638 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √638 the pattern is [25; 3, 1, 6, 2, 6, 1, 3, 50] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √638 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 − 638y² = 1. Its smallest solution in positive whole numbers is x = 42,283, y = 1,674.

√638 in geometry and everyday measurements

• A square garage floor of 638 square feet measures about 25.26 ft (25 ft 3 in) per side, and its corner-to-corner diagonal is √1276 ≈ 35.7 ft.
• 638 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √638 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 14 × 21 box, because 1² + 14² + 21² = 638.

Square roots near √638 and related roots

• The cube root of 638 is about 8.608753.
• Squaring undoes the root: (√638)² = 638, while 638² = 407,044 — the number whose square root is 638.

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