√394 at a glance
- Exact value
- √394
- Decimal (10 places)
- 19.8494332413
- Rounded
- 19.8 · 19.85 · 19.849
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.849433
- Prime factorization
- 2 × 197
- Cube root
- 7.331037
How to simplify √394
The prime factorization of 394 is 2 × 197. Every prime appears only once, so there is no pair to bring outside the radical — √394 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 394, 2 and 197 appear an odd number of times, so √394 is irrational and 19.8494332413 is a rounded value.
Where √394 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √394 lies between 19 and 20. 394 is 33 above 361 and 6 below 400, so the root is closer to 20.
- Straight line between 361 and 400: 19.8462 (0.02% low)
- Tangent from 19, i.e. 19 + 33 ÷ 38: 19.8684 (0.1% high)
- Tangent from 20, i.e. 20 − 6 ÷ 40: 19.8500 (0% high)
For √394 the tangent at 20 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 394 is just 6 below 400.
Finding √394 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, 20 (20² = 400):
| Step | Guess x | 394 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 19.7000000000 | 19.8500000000 | 3 |
| 2 | 19.8500000000 | 19.8488664987 | 19.8494332494 | 8 |
| 3 | 19.8494332494 | 19.8494332332 | 19.8494332413 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √394 = 19.8494332413 to every decimal shown.
√394 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √394 the pattern is [19; 1, 5, 1, 1, 1, 3, 1, 3, 5, 2, 2, 5, …] with the block of 21 terms after the semicolon repeating forever (only the first 12 of the 21 are shown). A pattern that never ends is one more proof that √394 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 |
|---|---|---|
| 19/1 | 19.0000000000 | 8.5 × 10⁻¹ |
| 20/1 | 20.0000000000 | 1.5 × 10⁻¹ |
| 119/6 | 19.8333333333 | 1.6 × 10⁻² |
| 139/7 | 19.8571428571 | 7.7 × 10⁻³ |
| 258/13 | 19.8461538462 | 3.3 × 10⁻³ |
| 397/20 | 19.8500000000 | 5.7 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 394y² = 1. Its smallest solution in positive whole numbers is x = 312,086,396,361,222,451, y = 15,722,685,507,826,110 — 18 digits for x, even though 394 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 395,023,035² − 394 × 19,900,973² = −1.
√394 in geometry and everyday measurements
- A square patio or deck of 394 square feet is about 19.85 ft (19 ft 10 in) on each side, so edging all the way around takes 4 × √394 ≈ 79.4 ft.
- 394 = 13² + 15², so by the Pythagorean theorem √394 is the diagonal of a 13 × 15 rectangle — and the distance between the points (0, 0) and (13, 15) on a grid.
Square roots near √394 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √391 | √391 | 19.7737 | No |
| √392 | 14√2 | 19.7990 | No |
| √393 | √393 | 19.8242 | No |
| √394 | √394 | 19.8494 | No |
| √395 | √395 | 19.8746 | No |
| √396 | 6√11 | 19.8997 | No |
| √397 | √397 | 19.9249 | No |
- The cube root of 394 is about 7.331037.
- Squaring undoes the root: (√394)² = 394, while 394² = 155,236 — the number whose square root is 394.
Frequently asked questions
What is the square root of 394?
The square root of 394 is √394, about 19.8494332413. The negative root, −19.849433, also squares to 394.
Is the square root of 394 rational or irrational?
Irrational. 394 is not a perfect square — it falls between 361 and 400 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √394 be simplified?
No. 394 = 2 × 197 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √394 rounded to two decimal places?
√394 ≈ 19.85 to two decimal places (19.8 to one, 19.849 to three). Check: 19.85² = 394.0225, close to 394.