Square Root of 992

√992 at a glance

Exact value

4√62

Decimal (10 places)

31.4960314960

Rounded

31.5 · 31.50 · 31.496

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.496031

Prime factorization

2⁵ × 31

Cube root

9.973262

How to simplify √992

Look for the largest perfect square that divides 992. Here it is 16 (4²), because 992 = 16 × 62 and 62 has no square factor left:

The prime factorization tells the same story: 992 = 2⁵ × 31. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 2 × 31 stays inside.

992 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √992 = 2√248, and √248 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√62)² = 4² × 62 = 16 × 62 = 992. As a decimal, 4√62 = 4 × 7.874007874 ≈ 31.4960314960.

Where √992 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √992 lies between 31 and 32. 992 is 31 above 961 and 32 below 1,024, so the root is closer to 31.

• Straight line between 961 and 1,024: 31.4921 (0.01% low)
• Tangent from 31, i.e. 31 + 31 ÷ 62: 31.5000 (0.01% high)
• Tangent from 32, i.e. 32 − 32 ÷ 64: 31.5000 (0.01% high)

For √992 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √992 with the Babylonian method

Picture a rectangle with an area of 992 and one side x; the other side must be 992 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √992.

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 √992 = 31.4960314960 to every decimal shown.

√992 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √992 the pattern is [31; 2, 62] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √992 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 − 992y² = 1. Its smallest solution in positive whole numbers is x = 63, y = 2.

√992 in geometry and everyday measurements

• 992 square feet is 92.2 m². Laid out as a square — a small house footprint or a lot — it is about 31.5 ft (31 ft 6 in) on a side.
• 992 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √992 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 20 × 24 box, because 4² + 20² + 24² = 992.
• Since √992 = 4√62, a length of √992 is exactly 4 copies of the length √62 laid end to end.

Square roots near √992 and related roots

• The cube root of 992 is about 9.973262.
• Because 992 = 4 × 248, the root is twice √248: 2 × 15.748016 ≈ 31.496031.

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