Square Root of 656

√656 at a glance

Exact value

4√41

Decimal (10 places)

25.6124969497

Rounded

25.6 · 25.61 · 25.612

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.612497

Prime factorization

2⁴ × 41

Cube root

8.688963

How to simplify √656

Look for the largest perfect square that divides 656. Here it is 16 (4²), because 656 = 16 × 41 and 41 has no square factor left:

The prime factorization tells the same story: 656 = 2⁴ × 41. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 41 stays inside.

656 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √656 = 2√164, and √164 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√41)² = 4² × 41 = 16 × 41 = 656. As a decimal, 4√41 = 4 × 6.4031242374 ≈ 25.6124969497.

Where √656 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √656 lies between 25 and 26. 656 is 31 above 625 and 20 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.6078 (0.02% low)
• Tangent from 25, i.e. 25 + 31 ÷ 50: 25.6200 (0.03% high)
• Tangent from 26, i.e. 26 − 20 ÷ 52: 25.6154 (0.01% high)

For √656 the tangent at 26 wins, missing by only 0.0029. Tangent estimates shine when the number sits close to a perfect square — here 656 is just 20 below 676.

Finding √656 with the Babylonian method

Picture a rectangle with an area of 656 and one side x; the other side must be 656 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √656.

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 √656 = 25.6124969497 to every decimal shown.

√656 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √656 the pattern is [25; 1, 1, 1, 1, 2, 1, 1, 1, 1, 50] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √656 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 − 656y² = 1. Its smallest solution in positive whole numbers is x = 2,049, y = 80.

√656 in geometry and everyday measurements

• 656 square feet is 60.9 m². Laid out as a square — a small house footprint or a lot — it is about 25.61 ft (25 ft 7 in) on a side.
• 656 = 16² + 20², so by the Pythagorean theorem √656 is the diagonal of a 16 × 20 rectangle — and the distance between the points (0, 0) and (16, 20) on a grid.
• Since √656 = 4√41, a length of √656 is exactly 4 copies of the length √41 laid end to end.

Square roots near √656 and related roots

• The cube root of 656 is about 8.688963.
• Because 656 = 4 × 164, the root is twice √164: 2 × 12.806248 ≈ 25.612497.

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