Square Root of 341

√341 at a glance

Exact value

√341

Decimal (10 places)

18.4661853126

Rounded

18.5 · 18.47 · 18.466

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.466185

Prime factorization

11 × 31

Cube root

6.986368

How to simplify √341

The prime factorization of 341 is 11 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √341 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 341, 11 and 31 appear an odd number of times, so √341 is irrational and 18.4661853126 is a rounded value.

Where √341 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √341 lies between 18 and 19. 341 is 17 above 324 and 20 below 361, so the root is closer to 18.

• Straight line between 324 and 361: 18.4595 (0.04% low)
• Tangent from 18, i.e. 18 + 17 ÷ 36: 18.4722 (0.03% high)
• Tangent from 19, i.e. 19 − 20 ÷ 38: 18.4737 (0.04% high)

For √341 the tangent at 18 wins, missing by only 0.006. Tangent estimates shine when the number sits close to a perfect square — here 341 is just 17 above 324.

Finding √341 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 341: following the tangent line down to zero simplifies to averaging x with 341 ÷ x.

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 √341 = 18.4661853126 to every decimal shown.

√341 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √341 the pattern is [18; 2, 6, 1, 8, 2, 1, 2, 1, 2, 8, 1, 6, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √341 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 − 341y² = 1. Its smallest solution in positive whole numbers is x = 10,626,551, y = 575,460.

√341 in geometry and everyday measurements

• A square patio or deck of 341 square feet is about 18.47 ft (18 ft 6 in) on each side, so edging all the way around takes 4 × √341 ≈ 73.9 ft.
• 341 is not a sum of two whole-number squares — the prime factor 11 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √341 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 18 box, because 1² + 4² + 18² = 341.

Square roots near √341 and related roots

• The cube root of 341 is about 6.986368.
• Squaring undoes the root: (√341)² = 341, while 341² = 116,281 — the number whose square root is 341.

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