Square Root of 338

√338 at a glance

Exact value

13√2

Decimal (10 places)

18.3847763109

Rounded

18.4 · 18.38 · 18.385

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.384776

Prime factorization

2 × 13²

Cube root

6.965820

How to simplify √338

Look for the largest perfect square that divides 338. Here it is 169 (13²), because 338 = 169 × 2 and 2 has no square factor left:

The prime factorization tells the same story: 338 = 2 × 13². Each pair of equal primes leaves the radical as one factor, so 13 comes out and 2 stays inside.

Check: (13√2)² = 13² × 2 = 169 × 2 = 338. As a decimal, 13√2 = 13 × 1.4142135624 ≈ 18.3847763109.

Where √338 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √338 lies between 18 and 19. 338 is 14 above 324 and 23 below 361, so the root is closer to 18.

• Straight line between 324 and 361: 18.3784 (0.03% low)
• Tangent from 18, i.e. 18 + 14 ÷ 36: 18.3889 (0.02% high)
• Tangent from 19, i.e. 19 − 23 ÷ 38: 18.3947 (0.05% high)

For √338 the tangent at 18 wins, missing by only 0.0041. Tangent estimates shine when the number sits close to a perfect square — here 338 is just 14 above 324.

Finding √338 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 √338 = 18.3847763109 to every decimal shown.

√338 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √338 the pattern is [18; 2, 1, 1, 2, 36] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √338 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 − 338y² = 1. Its smallest solution in positive whole numbers is x = 114,243, y = 6,214. Because the period is odd, the equation with −1 on the right also has a solution: 239² − 338 × 13² = −1.

√338 in geometry and everyday measurements

• A square patio or deck of 338 square feet is about 18.38 ft (18 ft 5 in) on each side, so edging all the way around takes 4 × √338 ≈ 73.5 ft.
• 338 = 7² + 17² = 13² + 13², so by the Pythagorean theorem √338 is the diagonal of rectangles measuring 7 × 17 and 13 × 13 — and the distance between the points (0, 0) and (7, 17) on a grid.
• Since √338 = 13√2, a length of √338 is exactly 13 copies of the length √2 laid end to end.

Square roots near √338 and related roots

• The cube root of 338 is about 6.965820.
• Squaring undoes the root: (√338)² = 338, while 338² = 114,244 — the number whose square root is 338.

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