Square Root of 926

√926 at a glance

Exact value

√926

Decimal (10 places)

30.4302481094

Rounded

30.4 · 30.43 · 30.430

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.430248

Prime factorization

2 × 463

Cube root

9.746986

How to simplify √926

The prime factorization of 926 is 2 × 463. Every prime appears only once, so there is no pair to bring outside the radical — √926 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 926, 2 and 463 appear an odd number of times, so √926 is irrational and 30.4302481094 is a rounded value.

Where √926 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √926 lies between 30 and 31. 926 is 26 above 900 and 35 below 961, so the root is closer to 30.

• Straight line between 900 and 961: 30.4262 (0.01% low)
• Tangent from 30, i.e. 30 + 26 ÷ 60: 30.4333 (0.01% high)
• Tangent from 31, i.e. 31 − 35 ÷ 62: 30.4355 (0.02% high)

For √926 the tangent at 30 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 926 is just 26 above 900.

Finding √926 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, 30 (30² = 900):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √926 = 30.4302481094 to every decimal shown.

√926 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √926 the pattern is [30; 2, 3, 11, 1, 7, 1, 3, 2, 5, 1, 1, 1, …] with the block of 28 terms after the semicolon repeating forever (only the first 12 of the 28 are shown). A pattern that never ends is one more proof that √926 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 − 926y² = 1. Its smallest solution in positive whole numbers is x = 304,560,297,142,335, y = 10,008,472,361,032 — 15 digits for x, even though 926 is small, which is what makes Pell’s equation famous.

√926 in geometry and everyday measurements

• 926 square feet is 86 m². Laid out as a square — a small house footprint or a lot — it is about 30.43 ft (30 ft 5 in) on a side.
• 926 is not a sum of two whole-number squares — the prime factor 463 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √926 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 30 box, because 1² + 5² + 30² = 926.

Square roots near √926 and related roots

• The cube root of 926 is about 9.746986.
• Squaring undoes the root: (√926)² = 926, while 926² = 857,476 — the number whose square root is 926.

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