Square Root of 344

√344 at a glance

Exact value

2√86

Decimal (10 places)

18.5472369910

Rounded

18.5 · 18.55 · 18.547

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.547237

Prime factorization

2³ × 43

Cube root

7.006796

How to simplify √344

Look for the largest perfect square that divides 344. Here it is 4 (2²), because 344 = 4 × 86 and 86 has no square factor left:

The prime factorization tells the same story: 344 = 2³ × 43. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 43 stays inside.

Check: (2√86)² = 2² × 86 = 4 × 86 = 344. As a decimal, 2√86 = 2 × 9.2736184955 ≈ 18.5472369910.

Where √344 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √344 lies between 18 and 19. 344 is 20 above 324 and 17 below 361, so the root is closer to 19.

• Straight line between 324 and 361: 18.5405 (0.04% low)
• Tangent from 18, i.e. 18 + 20 ÷ 36: 18.5556 (0.04% high)
• Tangent from 19, i.e. 19 − 17 ÷ 38: 18.5526 (0.03% high)

For √344 the tangent at 19 wins, missing by only 0.0054. Tangent estimates shine when the number sits close to a perfect square — here 344 is just 17 below 361.

Finding √344 with the Babylonian method

Picture a rectangle with an area of 344 and one side x; the other side must be 344 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √344.

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 √344 = 18.5472369910 to every decimal shown.

√344 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √344 the pattern is [18; 1, 1, 4, 1, 3, 1, 4, 1, 1, 36] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √344 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 − 344y² = 1. Its smallest solution in positive whole numbers is x = 10,405, y = 561.

√344 in geometry and everyday measurements

• A square patio or deck of 344 square feet is about 18.55 ft (18 ft 7 in) on each side, so edging all the way around takes 4 × √344 ≈ 74.2 ft.
• 344 is not a sum of two whole-number squares — the prime factor 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √344 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 18 box, because 2² + 4² + 18² = 344.
• Since √344 = 2√86, a length of √344 is exactly 2 copies of the length √86 laid end to end.

Square roots near √344 and related roots

• The cube root of 344 is about 7.006796.
• Because 344 = 4 × 86, the root is twice √86: 2 × 9.273618 ≈ 18.547237.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.