Square Root of 938

√938 at a glance

Exact value

√938

Decimal (10 places)

30.6267856622

Rounded

30.6 · 30.63 · 30.627

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.626786

Prime factorization

2 × 7 × 67

Cube root

9.788909

How to simplify √938

The prime factorization of 938 is 2 × 7 × 67. Every prime appears only once, so there is no pair to bring outside the radical — √938 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 938, 2, 7 and 67 appear an odd number of times, so √938 is irrational and 30.6267856622 is a rounded value.

Where √938 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √938 lies between 30 and 31. 938 is 38 above 900 and 23 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.6230 (0.01% low)
• Tangent from 30, i.e. 30 + 38 ÷ 60: 30.6333 (0.02% high)
• Tangent from 31, i.e. 31 − 23 ÷ 62: 30.6290 (0.01% high)

For √938 the tangent at 31 wins, missing by only 0.0022. Tangent estimates shine when the number sits close to a perfect square — here 938 is just 23 below 961.

Finding √938 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, 31 (31² = 961):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √938 = 30.6267856622 to every decimal shown.

√938 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √938 the pattern is [30; 1, 1, 1, 2, 8, 2, 1, 1, 1, 60] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √938 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 − 938y² = 1. Its smallest solution in positive whole numbers is x = 17,151, y = 560.

√938 in geometry and everyday measurements

• 938 square feet is 87.1 m². Laid out as a square — a small house footprint or a lot — it is about 30.63 ft (30 ft 8 in) on a side.
• 938 is not a sum of two whole-number squares — the prime factor 7 and 67 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √938 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 19 × 24 box, because 1² + 19² + 24² = 938.

Square roots near √938 and related roots

• The cube root of 938 is about 9.788909.
• Squaring undoes the root: (√938)² = 938, while 938² = 879,844 — the number whose square root is 938.

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