Square Root of 635

√635 at a glance

Exact value

√635

Decimal (10 places)

25.1992063367

Rounded

25.2 · 25.20 · 25.199

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.199206

Prime factorization

5 × 127

Cube root

8.595238

How to simplify √635

The prime factorization of 635 is 5 × 127. Every prime appears only once, so there is no pair to bring outside the radical — √635 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 635, 5 and 127 appear an odd number of times, so √635 is irrational and 25.1992063367 is a rounded value.

Where √635 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √635 lies between 25 and 26. 635 is 10 above 625 and 41 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.1961 (0.01% low)
• Tangent from 25, i.e. 25 + 10 ÷ 50: 25.2000 (0% high)
• Tangent from 26, i.e. 26 − 41 ÷ 52: 25.2115 (0.05% high)

For √635 the tangent at 25 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 635 is just 10 above 625.

Finding √635 with the Babylonian method

If a guess is too big, 635 ÷ 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√635) in one step.

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

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √635 = 25.1992063367 to every decimal shown.

√635 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √635 the pattern is [25; 5, 50] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √635 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 − 635y² = 1. Its smallest solution in positive whole numbers is x = 126, y = 5.

√635 in geometry and everyday measurements

• A square garage floor of 635 square feet measures about 25.2 ft (25 ft 2 in) per side, and its corner-to-corner diagonal is √1270 ≈ 35.6 ft.
• 635 is not a sum of two whole-number squares — the prime factor 127 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √635 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 25 box, because 1² + 3² + 25² = 635.

Square roots near √635 and related roots

• The cube root of 635 is about 8.595238.
• Squaring undoes the root: (√635)² = 635, while 635² = 403,225 — the number whose square root is 635.

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