Square Root of 658

√658 at a glance

Exact value

√658

Decimal (10 places)

25.6515106768

Rounded

25.7 · 25.65 · 25.652

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.651511

Prime factorization

2 × 7 × 47

Cube root

8.697784

How to simplify √658

The prime factorization of 658 is 2 × 7 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √658 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 658, 2, 7 and 47 appear an odd number of times, so √658 is irrational and 25.6515106768 is a rounded value.

Where √658 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √658 lies between 25 and 26. 658 is 33 above 625 and 18 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.6471 (0.02% low)
• Tangent from 25, i.e. 25 + 33 ÷ 50: 25.6600 (0.03% high)
• Tangent from 26, i.e. 26 − 18 ÷ 52: 25.6538 (0.01% high)

For √658 the tangent at 26 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 658 is just 18 below 676.

Finding √658 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, 26 (26² = 676):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √658 = 25.6515106768 to every decimal shown.

√658 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √658 the pattern is [25; 1, 1, 1, 6, 1, 1, 1, 50] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √658 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 − 658y² = 1. Its smallest solution in positive whole numbers is x = 1,693, y = 66.

√658 in geometry and everyday measurements

• 658 square feet is 61.1 m². Laid out as a square — a small house footprint or a lot — it is about 25.65 ft (25 ft 8 in) on a side.
• 658 is not a sum of two whole-number squares — the prime factor 7 and 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √658 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 24 box, because 1² + 9² + 24² = 658.

Square roots near √658 and related roots

• The cube root of 658 is about 8.697784.
• Squaring undoes the root: (√658)² = 658, while 658² = 432,964 — the number whose square root is 658.

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