Square Root of 655

√655 at a glance

Exact value

√655

Decimal (10 places)

25.5929677841

Rounded

25.6 · 25.59 · 25.593

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.592968

Prime factorization

5 × 131

Cube root

8.684546

How to simplify √655

The prime factorization of 655 is 5 × 131. Every prime appears only once, so there is no pair to bring outside the radical — √655 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 655, 5 and 131 appear an odd number of times, so √655 is irrational and 25.5929677841 is a rounded value.

Where √655 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √655 lies between 25 and 26. 655 is 30 above 625 and 21 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.5882 (0.02% low)
• Tangent from 25, i.e. 25 + 30 ÷ 50: 25.6000 (0.03% high)
• Tangent from 26, i.e. 26 − 21 ÷ 52: 25.5962 (0.01% high)

For √655 the tangent at 26 wins, missing by only 0.0032. Tangent estimates shine when the number sits close to a perfect square — here 655 is just 21 below 676.

Finding √655 with the Babylonian method

If a guess is too big, 655 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√655) in one step.

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 √655 = 25.5929677841 to every decimal shown.

√655 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √655 the pattern is [25; 1, 1, 2, 5, 3, 2, 8, 10, 8, 2, 3, 5, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √655 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 − 655y² = 1. Its smallest solution in positive whole numbers is x = 737,709,209, y = 28,824,684.

√655 in geometry and everyday measurements

• 655 square feet is 60.9 m². Laid out as a square — a small house footprint or a lot — it is about 25.59 ft (25 ft 7 in) on a side.
• 655 is not a sum of two whole-number squares — the prime factor 131 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √655 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √655 as its space diagonal.

Square roots near √655 and related roots

• The cube root of 655 is about 8.684546.
• Squaring undoes the root: (√655)² = 655, while 655² = 429,025 — the number whose square root is 655.

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