Square Root of 984

√984 at a glance

Exact value

2√246

Decimal (10 places)

31.3687742827

Rounded

31.4 · 31.37 · 31.369

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.368774

Prime factorization

2³ × 3 × 41

Cube root

9.946380

How to simplify √984

Look for the largest perfect square that divides 984. Here it is 4 (2²), because 984 = 4 × 246 and 246 has no square factor left:

The prime factorization tells the same story: 984 = 2³ × 3 × 41. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 3 × 41 stays inside.

Check: (2√246)² = 2² × 246 = 4 × 246 = 984. As a decimal, 2√246 = 2 × 15.6843871414 ≈ 31.3687742827.

Where √984 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √984 lies between 31 and 32. 984 is 23 above 961 and 40 below 1,024, so the root is closer to 31.

• Straight line between 961 and 1,024: 31.3651 (0.01% low)
• Tangent from 31, i.e. 31 + 23 ÷ 62: 31.3710 (0.01% high)
• Tangent from 32, i.e. 32 − 40 ÷ 64: 31.3750 (0.02% high)

For √984 the tangent at 31 wins, missing by only 0.0022. Tangent estimates shine when the number sits close to a perfect square — here 984 is just 23 above 961.

Finding √984 with the Babylonian method

Picture a rectangle with an area of 984 and one side x; the other side must be 984 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √984.

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 √984 = 31.3687742827 to every decimal shown.

√984 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √984 the pattern is [31; 2, 1, 2, 2, 7, 2, 2, 1, 2, 62] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √984 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 − 984y² = 1. Its smallest solution in positive whole numbers is x = 88,805, y = 2,831.

√984 in geometry and everyday measurements

• 984 square feet is 91.4 m². Laid out as a square — a small house footprint or a lot — it is about 31.37 ft (31 ft 4 in) on a side.
• 984 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 √984 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 14 × 28 box, because 2² + 14² + 28² = 984.
• Since √984 = 2√246, a length of √984 is exactly 2 copies of the length √246 laid end to end.

Square roots near √984 and related roots

• The cube root of 984 is about 9.946380.
• Because 984 = 4 × 246, the root is twice √246: 2 × 15.684387 ≈ 31.368774.

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