Square Root of 346

√346 at a glance

Exact value

√346

Decimal (10 places)

18.6010752377

Rounded

18.6 · 18.60 · 18.601

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.601075

Prime factorization

2 × 173

Cube root

7.020349

How to simplify √346

The prime factorization of 346 is 2 × 173. Every prime appears only once, so there is no pair to bring outside the radical — √346 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 346, 2 and 173 appear an odd number of times, so √346 is irrational and 18.6010752377 is a rounded value.

Where √346 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √346 lies between 18 and 19. 346 is 22 above 324 and 15 below 361, so the root is closer to 19.

• Straight line between 324 and 361: 18.5946 (0.03% low)
• Tangent from 18, i.e. 18 + 22 ÷ 36: 18.6111 (0.05% high)
• Tangent from 19, i.e. 19 − 15 ÷ 38: 18.6053 (0.02% high)

For √346 the tangent at 19 wins, missing by only 0.0042. Tangent estimates shine when the number sits close to a perfect square — here 346 is just 15 below 361.

Finding √346 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, 19 (19² = 361):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √346 = 18.6010752377 to every decimal shown.

√346 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √346 the pattern is [18; 1, 1, 1, 1, 36] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √346 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 − 346y² = 1. Its smallest solution in positive whole numbers is x = 17,299, y = 930. Because the period is odd, the equation with −1 on the right also has a solution: 93² − 346 × 5² = −1.

√346 in geometry and everyday measurements

• A square patio or deck of 346 square feet is about 18.6 ft (18 ft 7 in) on each side, so edging all the way around takes 4 × √346 ≈ 74.4 ft.
• 346 = 11² + 15², so by the Pythagorean theorem √346 is the diagonal of a 11 × 15 rectangle — and the distance between the points (0, 0) and (11, 15) on a grid.

Square roots near √346 and related roots

• The cube root of 346 is about 7.020349.
• Squaring undoes the root: (√346)² = 346, while 346² = 119,716 — the number whose square root is 346.

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