Square Root of 538

√538 at a glance

Exact value

√538

Decimal (10 places)

23.1948270095

Rounded

23.2 · 23.19 · 23.195

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.194827

Prime factorization

2 × 269

Cube root

8.133187

How to simplify √538

The prime factorization of 538 is 2 × 269. Every prime appears only once, so there is no pair to bring outside the radical — √538 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 538, 2 and 269 appear an odd number of times, so √538 is irrational and 23.1948270095 is a rounded value.

Where √538 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √538 lies between 23 and 24. 538 is 9 above 529 and 38 below 576, so the root is closer to 23.

• Straight line between 529 and 576: 23.1915 (0.01% low)
• Tangent from 23, i.e. 23 + 9 ÷ 46: 23.1957 (0% high)
• Tangent from 24, i.e. 24 − 38 ÷ 48: 23.2083 (0.06% high)

For √538 the tangent at 23 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 538 is just 9 above 529.

Finding √538 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, 23 (23² = 529):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √538 = 23.1948270095 to every decimal shown.

√538 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √538 the pattern is [23; 5, 7, 1, 1, 7, 5, 46] with the block of 7 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √538 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 − 538y² = 1. Its smallest solution in positive whole numbers is x = 9,536,081,203, y = 411,129,654. Because the period is odd, the equation with −1 on the right also has a solution: 69,051² − 538 × 2,977² = −1.

√538 in geometry and everyday measurements

• A square garage floor of 538 square feet measures about 23.19 ft (23 ft 2 in) per side, and its corner-to-corner diagonal is √1076 ≈ 32.8 ft.
• 538 = 3² + 23², so by the Pythagorean theorem √538 is the diagonal of a 3 × 23 rectangle — and the distance between the points (0, 0) and (3, 23) on a grid.

Square roots near √538 and related roots

• The cube root of 538 is about 8.133187.
• Squaring undoes the root: (√538)² = 538, while 538² = 289,444 — the number whose square root is 538.

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