Square Root of 271

√271 at a glance

Exact value

√271

Decimal (10 places)

16.4620776332

Rounded

16.5 · 16.46 · 16.462

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.462078

Prime factorization

271

Cube root

6.471274

How to simplify √271

271 is a prime number, so its only factors are 1 and 271. There is no perfect-square factor to pull out, which means √271 is already in its simplest radical form.

The square root of any prime is irrational. If √271 were a fraction a/b in lowest terms, then a² = 271b², so 271 would divide a — and then 271 would divide b too, contradicting “lowest terms.” That is why the decimal 16.4620776332 is only a rounded value.

Where √271 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √271 lies between 16 and 17. 271 is 15 above 256 and 18 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.4545 (0.05% low)
• Tangent from 16, i.e. 16 + 15 ÷ 32: 16.4688 (0.04% high)
• Tangent from 17, i.e. 17 − 18 ÷ 34: 16.4706 (0.05% high)

For √271 the tangent at 16 wins, missing by only 0.0067. Tangent estimates shine when the number sits close to a perfect square — here 271 is just 15 above 256.

Finding √271 with the Babylonian method

If a guess is too big, 271 ÷ 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√271) in one step.

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

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √271 = 16.4620776332 to every decimal shown.

√271 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √271 the pattern is [16; 2, 6, 10, 1, 4, 1, 1, 2, 1, 2, 1, 15, …] with the block of 24 terms after the semicolon repeating forever (only the first 12 of the 24 are shown). A pattern that never ends is one more proof that √271 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 − 271y² = 1. Its smallest solution in positive whole numbers is x = 115,974,983,600, y = 7,044,978,537.

√271 in geometry and everyday measurements

• A square patio or deck of 271 square feet is about 16.46 ft (16 ft 6 in) on each side, so edging all the way around takes 4 × √271 ≈ 65.8 ft.
• 271 is not a sum of two whole-number squares — 271 is itself a prime that is one less than a multiple of 4, which rules that out — so √271 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 √271 as its space diagonal.

Square roots near √271 and related roots

• The cube root of 271 is about 6.471274.
• Squaring undoes the root: (√271)² = 271, while 271² = 73,441 — the number whose square root is 271.

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