Square Root of 238

√238 at a glance

Exact value

√238

Decimal (10 places)

15.4272486205

Rounded

15.4 · 15.43 · 15.427

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.427249

Prime factorization

2 × 7 × 17

Cube root

6.197154

How to simplify √238

The prime factorization of 238 is 2 × 7 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √238 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 238, 2, 7 and 17 appear an odd number of times, so √238 is irrational and 15.4272486205 is a rounded value.

Where √238 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √238 lies between 15 and 16. 238 is 13 above 225 and 18 below 256, so the root is closer to 15.

• Straight line between 225 and 256: 15.4194 (0.05% low)
• Tangent from 15, i.e. 15 + 13 ÷ 30: 15.4333 (0.04% high)
• Tangent from 16, i.e. 16 − 18 ÷ 32: 15.4375 (0.07% high)

For √238 the tangent at 15 wins, missing by only 0.0061. Tangent estimates shine when the number sits close to a perfect square — here 238 is just 13 above 225.

Finding √238 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, 15 (15² = 225):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √238 = 15.4272486205 to every decimal shown.

√238 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √238 the pattern is [15; 2, 2, 1, 14, 1, 2, 2, 30] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √238 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 − 238y² = 1. Its smallest solution in positive whole numbers is x = 11,663, y = 756.

√238 in geometry and everyday measurements

• A square patio or deck of 238 square feet is about 15.43 ft (15 ft 5 in) on each side, so edging all the way around takes 4 × √238 ≈ 61.7 ft.
• 238 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √238 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 3 × 15 box, because 2² + 3² + 15² = 238.

Square roots near √238 and related roots

• The cube root of 238 is about 6.197154.
• Four times the radicand doubles the root: √952 = 2 × √238 ≈ 30.854497.

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