Square Root of 257

√257 at a glance

Exact value

√257

Decimal (10 places)

16.0312195419

Rounded

16.0 · 16.03 · 16.031

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.031220

Prime factorization

257

Cube root

6.357861

How to simplify √257

257 is a prime number, so its only factors are 1 and 257. There is no perfect-square factor to pull out, which means √257 is already in its simplest radical form.

The square root of any prime is irrational. If √257 were a fraction a/b in lowest terms, then a² = 257b², so 257 would divide a — and then 257 would divide b too, contradicting “lowest terms.” That is why the decimal 16.0312195419 is only a rounded value.

Where √257 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √257 lies between 16 and 17. 257 is 1 above 256 and 32 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.0303 (0.01% low)
• Tangent from 16, i.e. 16 + 1 ÷ 32: 16.0313 (0% high)
• Tangent from 17, i.e. 17 − 32 ÷ 34: 16.0588 (0.17% high)

For √257 the tangent at 16 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 257 is just 1 above 256.

Finding √257 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 257: following the tangent line down to zero simplifies to averaging x with 257 ÷ x.

Start from the nearest whole number, 16 (16² = 256):

Because the starting guess was already close, two steps are enough to match √257 = 16.0312195419 to every decimal shown.

√257 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √257 the pattern is [16; 32] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 257 is one more than a perfect square (16² + 1). A pattern that never ends is one more proof that √257 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 − 257y² = 1. Its smallest solution in positive whole numbers is x = 513, y = 32. Because the period is odd, the equation with −1 on the right also has a solution: 16² − 257 × 1² = −1.

√257 in geometry and everyday measurements

• A square patio or deck of 257 square feet is about 16.03 ft (16 ft) on each side, so edging all the way around takes 4 × √257 ≈ 64.1 ft.
• 257 = 1² + 16², so by the Pythagorean theorem √257 is the diagonal of a 1 × 16 rectangle — and the distance between the points (0, 0) and (1, 16) on a grid.

Square roots near √257 and related roots

• The cube root of 257 is about 6.357861.
• Squaring undoes the root: (√257)² = 257, while 257² = 66,049 — the number whose square root is 257.

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