Square Root of 637

√637 at a glance

Exact value

7√13

Decimal (10 places)

25.2388589282

Rounded

25.2 · 25.24 · 25.239

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.238859

Prime factorization

7² × 13

Cube root

8.604252

How to simplify √637

Look for the largest perfect square that divides 637. Here it is 49 (7²), because 637 = 49 × 13 and 13 has no square factor left:

The prime factorization tells the same story: 637 = 7² × 13. Each pair of equal primes leaves the radical as one factor, so 7 comes out and 13 stays inside.

Check: (7√13)² = 7² × 13 = 49 × 13 = 637. As a decimal, 7√13 = 7 × 3.6055512755 ≈ 25.2388589282.

Where √637 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √637 lies between 25 and 26. 637 is 12 above 625 and 39 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.2353 (0.01% low)
• Tangent from 25, i.e. 25 + 12 ÷ 50: 25.2400 (0% high)
• Tangent from 26, i.e. 26 − 39 ÷ 52: 25.2500 (0.04% high)

For √637 the tangent at 25 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 637 is just 12 above 625.

Finding √637 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 637: following the tangent line down to zero simplifies to averaging x with 637 ÷ x.

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 √637 = 25.2388589282 to every decimal shown.

√637 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √637 the pattern is [25; 4, 5, 2, 1, 3, 1, 1, 12, 16, 1, 2, 1, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √637 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 − 637y² = 1. Its smallest solution in positive whole numbers is x = 1,419,278,889,601, y = 56,233,877,040 — 13 digits for x, even though 637 is small, which is what makes Pell’s equation famous.

√637 in geometry and everyday measurements

• A square garage floor of 637 square feet measures about 25.24 ft (25 ft 3 in) per side, and its corner-to-corner diagonal is √1274 ≈ 35.7 ft.
• 637 = 14² + 21², so by the Pythagorean theorem √637 is the diagonal of a 14 × 21 rectangle — and the distance between the points (0, 0) and (14, 21) on a grid.
• Since √637 = 7√13, a length of √637 is exactly 7 copies of the length √13 laid end to end.

Square roots near √637 and related roots

• The cube root of 637 is about 8.604252.
• Squaring undoes the root: (√637)² = 637, while 637² = 405,769 — the number whose square root is 637.

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