Square Root of 931

√931 at a glance

Exact value

7√19

Decimal (10 places)

30.5122926048

Rounded

30.5 · 30.51 · 30.512

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.512293

Prime factorization

7² × 19

Cube root

9.764497

How to simplify √931

Look for the largest perfect square that divides 931. Here it is 49 (7²), because 931 = 49 × 19 and 19 has no square factor left:

The prime factorization tells the same story: 931 = 7² × 19. Each pair of equal primes leaves the radical as one factor, so 7 comes out and 19 stays inside.

Check: (7√19)² = 7² × 19 = 49 × 19 = 931. As a decimal, 7√19 = 7 × 4.3588989435 ≈ 30.5122926048.

Where √931 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √931 lies between 30 and 31. 931 is 31 above 900 and 30 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.5082 (0.01% low)
• Tangent from 30, i.e. 30 + 31 ÷ 60: 30.5167 (0.01% high)
• Tangent from 31, i.e. 31 − 30 ÷ 62: 30.5161 (0.01% high)

For √931 the tangent at 31 wins, missing by only 0.0038. Tangent estimates shine when the number sits close to a perfect square — here 931 is just 30 below 961.

Finding √931 with the Babylonian method

If a guess is too big, 931 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√931) in one step.

Start from the nearest whole number, 31 (31² = 961):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √931 = 30.5122926048 to every decimal shown.

√931 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √931 the pattern is [30; 1, 1, 19, 1, 5, 6, 1, 1, 1, 1, 2, 1, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √931 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 − 931y² = 1. Its smallest solution in positive whole numbers is x = 6,681,448,801, y = 218,975,640.

√931 in geometry and everyday measurements

• 931 square feet is 86.5 m². Laid out as a square — a small house footprint or a lot — it is about 30.51 ft (30 ft 6 in) on a side.
• 931 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √931 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 9 × 29 box, because 3² + 9² + 29² = 931.
• Since √931 = 7√19, a length of √931 is exactly 7 copies of the length √19 laid end to end.

Square roots near √931 and related roots

• The cube root of 931 is about 9.764497.
• Squaring undoes the root: (√931)² = 931, while 931² = 866,761 — the number whose square root is 931.

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