Square Root of 639

√639 at a glance

Exact value

3√71

Decimal (10 places)

25.2784493195

Rounded

25.3 · 25.28 · 25.278

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.278449

Prime factorization

3² × 71

Cube root

8.613248

How to simplify √639

Look for the largest perfect square that divides 639. Here it is 9 (3²), because 639 = 9 × 71 and 71 has no square factor left:

The prime factorization tells the same story: 639 = 3² × 71. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 71 stays inside.

Check: (3√71)² = 3² × 71 = 9 × 71 = 639. As a decimal, 3√71 = 3 × 8.4261497732 ≈ 25.2784493195.

Where √639 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √639 lies between 25 and 26. 639 is 14 above 625 and 37 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.2745 (0.02% low)
• Tangent from 25, i.e. 25 + 14 ÷ 50: 25.2800 (0.01% high)
• Tangent from 26, i.e. 26 − 37 ÷ 52: 25.2885 (0.04% high)

For √639 the tangent at 25 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 639 is just 14 above 625.

Finding √639 with the Babylonian method

If a guess is too big, 639 ÷ 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√639) in one step.

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 √639 = 25.2784493195 to every decimal shown.

√639 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √639 the pattern is [25; 3, 1, 1, 2, 4, 4, 1, 4, 1, 4, 4, 2, …] 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 √639 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 − 639y² = 1. Its smallest solution in positive whole numbers is x = 24,220,799, y = 958,160.

√639 in geometry and everyday measurements

• A square garage floor of 639 square feet measures about 25.28 ft (25 ft 3 in) per side, and its corner-to-corner diagonal is √1278 ≈ 35.7 ft.
• 639 is not a sum of two whole-number squares — the prime factor 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √639 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 √639 as its space diagonal.
• Since √639 = 3√71, a length of √639 is exactly 3 copies of the length √71 laid end to end.

Square roots near √639 and related roots

• The cube root of 639 is about 8.613248.
• Squaring undoes the root: (√639)² = 639, while 639² = 408,321 — the number whose square root is 639.

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