Square Root of 334

√334 at a glance

Exact value

√334

Decimal (10 places)

18.2756668825

Rounded

18.3 · 18.28 · 18.276

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.275667

Prime factorization

2 × 167

Cube root

6.938232

How to simplify √334

The prime factorization of 334 is 2 × 167. Every prime appears only once, so there is no pair to bring outside the radical — √334 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 334, 2 and 167 appear an odd number of times, so √334 is irrational and 18.2756668825 is a rounded value.

Where √334 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √334 lies between 18 and 19. 334 is 10 above 324 and 27 below 361, so the root is closer to 18.

• Straight line between 324 and 361: 18.2703 (0.03% low)
• Tangent from 18, i.e. 18 + 10 ÷ 36: 18.2778 (0.01% high)
• Tangent from 19, i.e. 19 − 27 ÷ 38: 18.2895 (0.08% high)

For √334 the tangent at 18 wins, missing by only 0.0021. Tangent estimates shine when the number sits close to a perfect square — here 334 is just 10 above 324.

Finding √334 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 18 (18² = 324):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √334 = 18.2756668825 to every decimal shown.

√334 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √334 the pattern is [18; 3, 1, 1, 1, 2, 5, 1, 2, 2, 11, 1, 3, …] with the block of 28 terms after the semicolon repeating forever (only the first 12 of the 28 are shown). A pattern that never ends is one more proof that √334 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 − 334y² = 1. Its smallest solution in positive whole numbers is x = 63,804,373,719,695, y = 3,491,219,999,244 — 14 digits for x, even though 334 is small, which is what makes Pell’s equation famous.

√334 in geometry and everyday measurements

• A square patio or deck of 334 square feet is about 18.28 ft (18 ft 3 in) on each side, so edging all the way around takes 4 × √334 ≈ 73.1 ft.
• 334 is not a sum of two whole-number squares — the prime factor 167 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √334 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 18 box, because 1² + 3² + 18² = 334.

Square roots near √334 and related roots

• The cube root of 334 is about 6.938232.
• Squaring undoes the root: (√334)² = 334, while 334² = 111,556 — the number whose square root is 334.

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