Square Root of 996

√996 at a glance

Exact value

2√249

Decimal (10 places)

31.5594676761

Rounded

31.6 · 31.56 · 31.559

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.559468

Prime factorization

2² × 3 × 83

Cube root

9.986649

How to simplify √996

Look for the largest perfect square that divides 996. Here it is 4 (2²), because 996 = 4 × 249 and 249 has no square factor left:

The prime factorization tells the same story: 996 = 2² × 3 × 83. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 83 stays inside.

Check: (2√249)² = 2² × 249 = 4 × 249 = 996. As a decimal, 2√249 = 2 × 15.7797338381 ≈ 31.5594676761.

Where √996 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √996 lies between 31 and 32. 996 is 35 above 961 and 28 below 1,024, so the root is closer to 32.

• Straight line between 961 and 1,024: 31.5556 (0.01% low)
• Tangent from 31, i.e. 31 + 35 ÷ 62: 31.5645 (0.02% high)
• Tangent from 32, i.e. 32 − 28 ÷ 64: 31.5625 (0.01% high)

For √996 the tangent at 32 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 996 is just 28 below 1,024.

Finding √996 with the Babylonian method

Picture a rectangle with an area of 996 and one side x; the other side must be 996 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √996.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √996 = 31.5594676761 to every decimal shown.

√996 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √996 the pattern is [31; 1, 1, 3, 1, 2, 2, 1, 1, 1, 4, 1, 1, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √996 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 − 996y² = 1. Its smallest solution in positive whole numbers is x = 8,553,815, y = 271,038.

√996 in geometry and everyday measurements

• 996 square feet is 92.5 m². Laid out as a square — a small house footprint or a lot — it is about 31.56 ft (31 ft 7 in) on a side.
• 996 is not a sum of two whole-number squares — the prime factor 3 and 83 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √996 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 14 × 28 box, because 4² + 14² + 28² = 996.
• Since √996 = 2√249, a length of √996 is exactly 2 copies of the length √249 laid end to end.

Square roots near √996 and related roots

• The cube root of 996 is about 9.986649.
• Because 996 = 4 × 249, the root is twice √249: 2 × 15.779734 ≈ 31.559468.

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