√385 at a glance
- Exact value
- √385
- Decimal (10 places)
- 19.6214168703
- Rounded
- 19.6 · 19.62 · 19.621
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.621417
- Prime factorization
- 5 × 7 × 11
- Cube root
- 7.274786
How to simplify √385
The prime factorization of 385 is 5 × 7 × 11. Every prime appears only once, so there is no pair to bring outside the radical — √385 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 385, 5, 7 and 11 appear an odd number of times, so √385 is irrational and 19.6214168703 is a rounded value.
Where √385 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √385 lies between 19 and 20. 385 is 24 above 361 and 15 below 400, so the root is closer to 20.
- Straight line between 361 and 400: 19.6154 (0.03% low)
- Tangent from 19, i.e. 19 + 24 ÷ 38: 19.6316 (0.05% high)
- Tangent from 20, i.e. 20 − 15 ÷ 40: 19.6250 (0.02% high)
For √385 the tangent at 20 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 385 is just 15 below 400.
Finding √385 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 385: following the tangent line down to zero simplifies to averaging x with 385 ÷ x.
Start from the nearest whole number, 20 (20² = 400):
| Step | Guess x | 385 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 19.2500000000 | 19.6250000000 | 2 |
| 2 | 19.6250000000 | 19.6178343949 | 19.6214171975 | 6 |
| 3 | 19.6214171975 | 19.6214165432 | 19.6214168703 | 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 √385 = 19.6214168703 to every decimal shown.
√385 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √385 the pattern is [19; 1, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √385 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 | 6.2 × 10⁻¹ |
| 20/1 | 20.0000000000 | 3.8 × 10⁻¹ |
| 39/2 | 19.5000000000 | 1.2 × 10⁻¹ |
| 59/3 | 19.6666666667 | 4.5 × 10⁻² |
| 98/5 | 19.6000000000 | 2.1 × 10⁻² |
| 157/8 | 19.6250000000 | 3.6 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 385y² = 1. Its smallest solution in positive whole numbers is x = 95,831, y = 4,884.
√385 in geometry and everyday measurements
- A square patio or deck of 385 square feet is about 19.62 ft (19 ft 7 in) on each side, so edging all the way around takes 4 × √385 ≈ 78.5 ft.
- 385 is not a sum of two whole-number squares — the prime factor 7 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √385 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 12 × 15 box, because 4² + 12² + 15² = 385.
Square roots near √385 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √382 | √382 | 19.5448 | No |
| √383 | √383 | 19.5704 | No |
| √384 | 8√6 | 19.5959 | No |
| √385 | √385 | 19.6214 | No |
| √386 | √386 | 19.6469 | No |
| √387 | 3√43 | 19.6723 | No |
| √388 | 2√97 | 19.6977 | No |
- The cube root of 385 is about 7.274786.
- Squaring undoes the root: (√385)² = 385, while 385² = 148,225 — the number whose square root is 385.
Frequently asked questions
What is the square root of 385?
The square root of 385 is √385, about 19.6214168703. The negative root, −19.621417, also squares to 385.
Is the square root of 385 rational or irrational?
Irrational. 385 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 √385 be simplified?
No. 385 = 5 × 7 × 11 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √385 rounded to two decimal places?
√385 ≈ 19.62 to two decimal places (19.6 to one, 19.621 to three). Check: 19.62² = 384.9444, close to 385.