Square Root of 234

√234 at a glance

Exact value

3√26

Decimal (10 places)

15.2970585408

Rounded

15.3 · 15.30 · 15.297

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.297059

Prime factorization

2 × 3² × 13

Cube root

6.162240

How to simplify √234

Look for the largest perfect square that divides 234. Here it is 9 (3²), because 234 = 9 × 26 and 26 has no square factor left:

The prime factorization tells the same story: 234 = 2 × 3² × 13. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 13 stays inside.

Check: (3√26)² = 3² × 26 = 9 × 26 = 234. As a decimal, 3√26 = 3 × 5.0990195136 ≈ 15.2970585408.

Where √234 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √234 lies between 15 and 16. 234 is 9 above 225 and 22 below 256, so the root is closer to 15.

• Straight line between 225 and 256: 15.2903 (0.04% low)
• Tangent from 15, i.e. 15 + 9 ÷ 30: 15.3000 (0.02% high)
• Tangent from 16, i.e. 16 − 22 ÷ 32: 15.3125 (0.1% high)

For √234 the tangent at 15 wins, missing by only 0.0029. Tangent estimates shine when the number sits close to a perfect square — here 234 is just 9 above 225.

Finding √234 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, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √234 = 15.2970585408 to every decimal shown.

√234 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √234 the pattern is [15; 3, 2, 1, 2, 1, 2, 3, 30] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √234 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 − 234y² = 1. Its smallest solution in positive whole numbers is x = 5,201, y = 340.

√234 in geometry and everyday measurements

• A square patio or deck of 234 square feet is about 15.3 ft (15 ft 4 in) on each side, so edging all the way around takes 4 × √234 ≈ 61.2 ft.
• 234 = 3² + 15², so by the Pythagorean theorem √234 is the diagonal of a 3 × 15 rectangle — and the distance between the points (0, 0) and (3, 15) on a grid.
• Since √234 = 3√26, a length of √234 is exactly 3 copies of the length √26 laid end to end.

Square roots near √234 and related roots

• The cube root of 234 is about 6.162240.
• Four times the radicand doubles the root: √936 = 2 × √234 ≈ 30.594117.

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