Square Root of 267

√267 at a glance

Exact value

√267

Decimal (10 places)

16.3401346384

Rounded

16.3 · 16.34 · 16.340

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.340135

Prime factorization

3 × 89

Cube root

6.439277

How to simplify √267

The prime factorization of 267 is 3 × 89. Every prime appears only once, so there is no pair to bring outside the radical — √267 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 267, 3 and 89 appear an odd number of times, so √267 is irrational and 16.3401346384 is a rounded value.

Where √267 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √267 lies between 16 and 17. 267 is 11 above 256 and 22 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.3333 (0.04% low)
• Tangent from 16, i.e. 16 + 11 ÷ 32: 16.3438 (0.02% high)
• Tangent from 17, i.e. 17 − 22 ÷ 34: 16.3529 (0.08% high)

For √267 the tangent at 16 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 267 is just 11 above 256.

Finding √267 with the Babylonian method

If a guess is too big, 267 ÷ 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√267) in one step.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √267 = 16.3401346384 to every decimal shown.

√267 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √267 the pattern is [16; 2, 1, 15, 1, 2, 32] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √267 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 − 267y² = 1. Its smallest solution in positive whole numbers is x = 2,402, y = 147.

√267 in geometry and everyday measurements

• A square patio or deck of 267 square feet is about 16.34 ft (16 ft 4 in) on each side, so edging all the way around takes 4 × √267 ≈ 65.4 ft.
• 267 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √267 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 11 × 11 box, because 5² + 11² + 11² = 267.

Square roots near √267 and related roots

• The cube root of 267 is about 6.439277.
• Squaring undoes the root: (√267)² = 267, while 267² = 71,289 — the number whose square root is 267.

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