Square Root of 349

√349 at a glance

Exact value

√349

Decimal (10 places)

18.6815416923

Rounded

18.7 · 18.68 · 18.682

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.681542

Prime factorization

349

Cube root

7.040581

How to simplify √349

349 is a prime number, so its only factors are 1 and 349. There is no perfect-square factor to pull out, which means √349 is already in its simplest radical form.

The square root of any prime is irrational. If √349 were a fraction a/b in lowest terms, then a² = 349b², so 349 would divide a — and then 349 would divide b too, contradicting “lowest terms.” That is why the decimal 18.6815416923 is only a rounded value.

Where √349 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √349 lies between 18 and 19. 349 is 25 above 324 and 12 below 361, so the root is closer to 19.

• Straight line between 324 and 361: 18.6757 (0.03% low)
• Tangent from 18, i.e. 18 + 25 ÷ 36: 18.6944 (0.07% high)
• Tangent from 19, i.e. 19 − 12 ÷ 38: 18.6842 (0.01% high)

For √349 the tangent at 19 wins, missing by only 0.0027. Tangent estimates shine when the number sits close to a perfect square — here 349 is just 12 below 361.

Finding √349 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 349: following the tangent line down to zero simplifies to averaging x with 349 ÷ x.

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 √349 = 18.6815416923 to every decimal shown.

√349 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √349 the pattern is [18; 1, 2, 7, 7, 2, 1, 36] with the block of 7 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √349 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 − 349y² = 1. Its smallest solution in positive whole numbers is x = 169,648,201, y = 9,081,060. Because the period is odd, the equation with −1 on the right also has a solution: 9,210² − 349 × 493² = −1.

√349 in geometry and everyday measurements

• A square patio or deck of 349 square feet is about 18.68 ft (18 ft 8 in) on each side, so edging all the way around takes 4 × √349 ≈ 74.7 ft.
• 349 = 5² + 18², so by the Pythagorean theorem √349 is the diagonal of a 5 × 18 rectangle — and the distance between the points (0, 0) and (5, 18) on a grid.

Square roots near √349 and related roots

• The cube root of 349 is about 7.040581.
• Squaring undoes the root: (√349)² = 349, while 349² = 121,801 — the number whose square root is 349.

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