Square Root of 999

√999 at a glance

Exact value

3√111

Decimal (10 places)

31.6069612586

Rounded

31.6 · 31.61 · 31.607

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.606961

Prime factorization

3³ × 37

Cube root

9.996666

How to simplify √999

Look for the largest perfect square that divides 999. Here it is 9 (3²), because 999 = 9 × 111 and 111 has no square factor left:

The prime factorization tells the same story: 999 = 3³ × 37. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 3 × 37 stays inside.

Check: (3√111)² = 3² × 111 = 9 × 111 = 999. As a decimal, 3√111 = 3 × 10.5356537529 ≈ 31.6069612586.

Where √999 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √999 lies between 31 and 32. 999 is 38 above 961 and 25 below 1,024, so the root is closer to 32.

• Straight line between 961 and 1,024: 31.6032 (0.01% low)
• Tangent from 31, i.e. 31 + 38 ÷ 62: 31.6129 (0.02% high)
• Tangent from 32, i.e. 32 − 25 ÷ 64: 31.6094 (0.01% high)

For √999 the tangent at 32 wins, missing by only 0.0024. Tangent estimates shine when the number sits close to a perfect square — here 999 is just 25 below 1,024.

Finding √999 with the Babylonian method

If a guess is too big, 999 ÷ 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√999) in one step.

Start from the nearest whole number, 32 (32² = 1,024):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √999 = 31.6069612586 to every decimal shown.

√999 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √999 the pattern is [31; 1, 1, 1, 1, 5, 6, 1, 5, 2, 5, 1, 6, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √999 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 − 999y² = 1. Its smallest solution in positive whole numbers is x = 102,688,615, y = 3,248,924.

√999 in geometry and everyday measurements

• 999 square feet is 92.8 m². Laid out as a square — a small house footprint or a lot — it is about 31.61 ft (31 ft 7 in) on a side.
• 999 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √999 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √999 as its space diagonal.
• Since √999 = 3√111, a length of √999 is exactly 3 copies of the length √111 laid end to end.

Square roots near √999 and related roots

• The cube root of 999 is about 9.996666.
• Squaring undoes the root: (√999)² = 999, while 999² = 998,001 — the number whose square root is 999.

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