√382 at a glance
- Exact value
- √382
- Decimal (10 places)
- 19.5448202857
- Rounded
- 19.5 · 19.54 · 19.545
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.544820
- Prime factorization
- 2 × 191
- Cube root
- 7.255842
How to simplify √382
The prime factorization of 382 is 2 × 191. Every prime appears only once, so there is no pair to bring outside the radical — √382 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 382, 2 and 191 appear an odd number of times, so √382 is irrational and 19.5448202857 is a rounded value.
Where √382 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √382 lies between 19 and 20. 382 is 21 above 361 and 18 below 400, so the root is closer to 20.
- Straight line between 361 and 400: 19.5385 (0.03% low)
- Tangent from 19, i.e. 19 + 21 ÷ 38: 19.5526 (0.04% high)
- Tangent from 20, i.e. 20 − 18 ÷ 40: 19.5500 (0.03% high)
For √382 the tangent at 20 wins, missing by only 0.0052. Tangent estimates shine when the number sits close to a perfect square — here 382 is just 18 below 400.
Finding √382 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 | 382 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 19.1000000000 | 19.5500000000 | 2 |
| 2 | 19.5500000000 | 19.5396419437 | 19.5448209719 | 6 |
| 3 | 19.5448209719 | 19.5448195995 | 19.5448202857 | 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 √382 = 19.5448202857 to every decimal shown.
√382 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √382 the pattern is [19; 1, 1, 5, 12, 1, 5, 1, 1, 2, 3, 1, 18, …] with the block of 24 terms after the semicolon repeating forever (only the first 12 of the 24 are shown). A pattern that never ends is one more proof that √382 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 | 5.4 × 10⁻¹ |
| 20/1 | 20.0000000000 | 4.6 × 10⁻¹ |
| 39/2 | 19.5000000000 | 4.5 × 10⁻² |
| 215/11 | 19.5454545455 | 6.3 × 10⁻⁴ |
| 2,619/134 | 19.5447761194 | 4.4 × 10⁻⁵ |
| 2,834/145 | 19.5448275862 | 7.3 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 382y² = 1. Its smallest solution in positive whole numbers is x = 164,998,439,999, y = 8,442,054,600.
√382 in geometry and everyday measurements
- A square patio or deck of 382 square feet is about 19.54 ft (19 ft 7 in) on each side, so edging all the way around takes 4 × √382 ≈ 78.2 ft.
- 382 is not a sum of two whole-number squares — the prime factor 191 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √382 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 7 × 18 box, because 3² + 7² + 18² = 382.
Square roots near √382 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √379 | √379 | 19.4679 | No |
| √380 | 2√95 | 19.4936 | No |
| √381 | √381 | 19.5192 | No |
| √382 | √382 | 19.5448 | No |
| √383 | √383 | 19.5704 | No |
| √384 | 8√6 | 19.5959 | No |
| √385 | √385 | 19.6214 | No |
- The cube root of 382 is about 7.255842.
- Squaring undoes the root: (√382)² = 382, while 382² = 145,924 — the number whose square root is 382.
Frequently asked questions
What is the square root of 382?
The square root of 382 is √382, about 19.5448202857. The negative root, −19.544820, also squares to 382.
Is the square root of 382 rational or irrational?
Irrational. 382 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 √382 be simplified?
No. 382 = 2 × 191 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √382 rounded to two decimal places?
√382 ≈ 19.54 to two decimal places (19.5 to one, 19.545 to three). Check: 19.54² = 381.8116, close to 382.