√381 at a glance
- Exact value
- √381
- Decimal (10 places)
- 19.5192212959
- Rounded
- 19.5 · 19.52 · 19.519
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.519221
- Prime factorization
- 3 × 127
- Cube root
- 7.249505
How to simplify √381
The prime factorization of 381 is 3 × 127. Every prime appears only once, so there is no pair to bring outside the radical — √381 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 381, 3 and 127 appear an odd number of times, so √381 is irrational and 19.5192212959 is a rounded value.
Where √381 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √381 lies between 19 and 20. 381 is 20 above 361 and 19 below 400, so the root is closer to 20.
- Straight line between 361 and 400: 19.5128 (0.03% low)
- Tangent from 19, i.e. 19 + 20 ÷ 38: 19.5263 (0.04% high)
- Tangent from 20, i.e. 20 − 19 ÷ 40: 19.5250 (0.03% high)
For √381 the tangent at 20 wins, missing by only 0.0058. Tangent estimates shine when the number sits close to a perfect square — here 381 is just 19 below 400.
Finding √381 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 381: following the tangent line down to zero simplifies to averaging x with 381 ÷ x.
Start from the nearest whole number, 20 (20² = 400):
| Step | Guess x | 381 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 19.0500000000 | 19.5250000000 | 2 |
| 2 | 19.5250000000 | 19.5134443022 | 19.5192221511 | 6 |
| 3 | 19.5192221511 | 19.5192204408 | 19.5192212959 | 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 √381 = 19.5192212959 to every decimal shown.
√381 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √381 the pattern is [19; 1, 1, 12, 1, 1, 38] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √381 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 | 5.2 × 10⁻¹ |
| 20/1 | 20.0000000000 | 4.8 × 10⁻¹ |
| 39/2 | 19.5000000000 | 1.9 × 10⁻² |
| 488/25 | 19.5200000000 | 7.8 × 10⁻⁴ |
| 527/27 | 19.5185185185 | 7.0 × 10⁻⁴ |
| 1,015/52 | 19.5192307692 | 9.5 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 381y² = 1. Its smallest solution in positive whole numbers is x = 1,015, y = 52.
√381 in geometry and everyday measurements
- A square patio or deck of 381 square feet is about 19.52 ft (19 ft 6 in) on each side, so edging all the way around takes 4 × √381 ≈ 78.1 ft.
- 381 is not a sum of two whole-number squares — the prime factor 3 and 127 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √381 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 19 box, because 2² + 4² + 19² = 381.
Square roots near √381 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √378 | 3√42 | 19.4422 | No |
| √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 |
- The cube root of 381 is about 7.249505.
- Squaring undoes the root: (√381)² = 381, while 381² = 145,161 — the number whose square root is 381.
Frequently asked questions
What is the square root of 381?
The square root of 381 is √381, about 19.5192212959. The negative root, −19.519221, also squares to 381.
Is the square root of 381 rational or irrational?
Irrational. 381 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 √381 be simplified?
No. 381 = 3 × 127 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √381 rounded to two decimal places?
√381 ≈ 19.52 to two decimal places (19.5 to one, 19.519 to three). Check: 19.52² = 381.0304, close to 381.