Square Root of 657

√657 at a glance

Exact value

3√73

Decimal (10 places)

25.6320112360

Rounded

25.6 · 25.63 · 25.632

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.632011

Prime factorization

3² × 73

Cube root

8.693376

How to simplify √657

Look for the largest perfect square that divides 657. Here it is 9 (3²), because 657 = 9 × 73 and 73 has no square factor left:

The prime factorization tells the same story: 657 = 3² × 73. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 73 stays inside.

Check: (3√73)² = 3² × 73 = 9 × 73 = 657. As a decimal, 3√73 = 3 × 8.5440037453 ≈ 25.6320112360.

Where √657 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √657 lies between 25 and 26. 657 is 32 above 625 and 19 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.6275 (0.02% low)
• Tangent from 25, i.e. 25 + 32 ÷ 50: 25.6400 (0.03% high)
• Tangent from 26, i.e. 26 − 19 ÷ 52: 25.6346 (0.01% high)

For √657 the tangent at 26 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 657 is just 19 below 676.

Finding √657 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 657: following the tangent line down to zero simplifies to averaging x with 657 ÷ x.

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 √657 = 25.6320112360 to every decimal shown.

√657 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √657 the pattern is [25; 1, 1, 1, 2, 1, 1, 5, 1, 4, 1, 5, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √657 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 − 657y² = 1. Its smallest solution in positive whole numbers is x = 2,281,249, y = 89,000.

√657 in geometry and everyday measurements

• 657 square feet is 61 m². Laid out as a square — a small house footprint or a lot — it is about 25.63 ft (25 ft 8 in) on a side.
• 657 = 9² + 24², so by the Pythagorean theorem √657 is the diagonal of a 9 × 24 rectangle — and the distance between the points (0, 0) and (9, 24) on a grid.
• Since √657 = 3√73, a length of √657 is exactly 3 copies of the length √73 laid end to end.

Square roots near √657 and related roots

• The cube root of 657 is about 8.693376.
• Squaring undoes the root: (√657)² = 657, while 657² = 431,649 — the number whose square root is 657.

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