√994 at a glance
- Exact value
- √994
- Decimal (10 places)
- 31.5277655409
- Rounded
- 31.5 · 31.53 · 31.528
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.527766
- Prime factorization
- 2 × 7 × 71
- Cube root
- 9.979960
How to simplify √994
The prime factorization of 994 is 2 × 7 × 71. Every prime appears only once, so there is no pair to bring outside the radical — √994 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 994, 2, 7 and 71 appear an odd number of times, so √994 is irrational and 31.5277655409 is a rounded value.
Where √994 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √994 lies between 31 and 32. 994 is 33 above 961 and 30 below 1,024, so the root is closer to 32.
- Straight line between 961 and 1,024: 31.5238 (0.01% low)
- Tangent from 31, i.e. 31 + 33 ÷ 62: 31.5323 (0.01% high)
- Tangent from 32, i.e. 32 − 30 ÷ 64: 31.5313 (0.01% high)
For √994 the tangent at 32 wins, missing by only 0.0035. Tangent estimates shine when the number sits close to a perfect square — here 994 is just 30 below 1,024.
Finding √994 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 32 (32² = 1,024):
| Step | Guess x | 994 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 32.0000000000 | 31.0625000000 | 31.5312500000 | 2 |
| 2 | 31.5312500000 | 31.5242814668 | 31.5277657334 | 6 |
| 3 | 31.5277657334 | 31.5277653483 | 31.5277655409 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √994 = 31.5277655409 to every decimal shown.
√994 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √994 the pattern is [31; 1, 1, 8, 1, 1, 62] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √994 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 31/1 | 31.0000000000 | 5.3 × 10⁻¹ |
| 32/1 | 32.0000000000 | 4.7 × 10⁻¹ |
| 63/2 | 31.5000000000 | 2.8 × 10⁻² |
| 536/17 | 31.5294117647 | 1.6 × 10⁻³ |
| 599/19 | 31.5263157895 | 1.4 × 10⁻³ |
| 1,135/36 | 31.5277777778 | 1.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 994y² = 1. Its smallest solution in positive whole numbers is x = 1,135, y = 36.
√994 in geometry and everyday measurements
- 994 square feet is 92.3 m². Laid out as a square — a small house footprint or a lot — it is about 31.53 ft (31 ft 6 in) on a side.
- 994 is not a sum of two whole-number squares — the prime factor 7 and 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √994 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 12 × 29 box, because 3² + 12² + 29² = 994.
Square roots near √994 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √991 | √991 | 31.4802 | No |
| √992 | 4√62 | 31.4960 | No |
| √993 | √993 | 31.5119 | No |
| √994 | √994 | 31.5278 | No |
| √995 | √995 | 31.5436 | No |
| √996 | 2√249 | 31.5595 | No |
| √997 | √997 | 31.5753 | No |
- The cube root of 994 is about 9.979960.
- Squaring undoes the root: (√994)² = 994, while 994² = 988,036 — the number whose square root is 994.
Frequently asked questions
What is the square root of 994?
The square root of 994 is √994, about 31.5277655409. The negative root, −31.527766, also squares to 994.
Is the square root of 994 rational or irrational?
Irrational. 994 is not a perfect square — it falls between 961 and 1024 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √994 be simplified?
No. 994 = 2 × 7 × 71 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √994 rounded to two decimal places?
√994 ≈ 31.53 to two decimal places (31.5 to one, 31.528 to three). Check: 31.53² = 994.1409, close to 994.