√393 at a glance
- Exact value
- √393
- Decimal (10 places)
- 19.8242276016
- Rounded
- 19.8 · 19.82 · 19.824
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.824228
- Prime factorization
- 3 × 131
- Cube root
- 7.324829
How to simplify √393
The prime factorization of 393 is 3 × 131. Every prime appears only once, so there is no pair to bring outside the radical — √393 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 393, 3 and 131 appear an odd number of times, so √393 is irrational and 19.8242276016 is a rounded value.
Where √393 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √393 lies between 19 and 20. 393 is 32 above 361 and 7 below 400, so the root is closer to 20.
- Straight line between 361 and 400: 19.8205 (0.02% low)
- Tangent from 19, i.e. 19 + 32 ÷ 38: 19.8421 (0.09% high)
- Tangent from 20, i.e. 20 − 7 ÷ 40: 19.8250 (0% high)
For √393 the tangent at 20 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 393 is just 7 below 400.
Finding √393 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 393: following the tangent line down to zero simplifies to averaging x with 393 ÷ x.
Start from the nearest whole number, 20 (20² = 400):
| Step | Guess x | 393 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 19.6500000000 | 19.8250000000 | 3 |
| 2 | 19.8250000000 | 19.8234552333 | 19.8242276166 | 7 |
| 3 | 19.8242276166 | 19.8242275866 | 19.8242276016 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √393 = 19.8242276016 to every decimal shown.
√393 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √393 the pattern is [19; 1, 4, 1, 2, 4, 1, 1, 1, 1, 12, 1, 1, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √393 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 | 8.2 × 10⁻¹ |
| 20/1 | 20.0000000000 | 1.8 × 10⁻¹ |
| 99/5 | 19.8000000000 | 2.4 × 10⁻² |
| 119/6 | 19.8333333333 | 9.1 × 10⁻³ |
| 337/17 | 19.8235294118 | 7.0 × 10⁻⁴ |
| 1,467/74 | 19.8243243243 | 9.7 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 393y² = 1. Its smallest solution in positive whole numbers is x = 46,437,143, y = 2,342,444.
√393 in geometry and everyday measurements
- A square patio or deck of 393 square feet is about 19.82 ft (19 ft 10 in) on each side, so edging all the way around takes 4 × √393 ≈ 79.3 ft.
- 393 is not a sum of two whole-number squares — the prime factor 3 and 131 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √393 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 14 × 14 box, because 1² + 14² + 14² = 393.
Square roots near √393 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √390 | √390 | 19.7484 | No |
| √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 |
- The cube root of 393 is about 7.324829.
- Squaring undoes the root: (√393)² = 393, while 393² = 154,449 — the number whose square root is 393.
Frequently asked questions
What is the square root of 393?
The square root of 393 is √393, about 19.8242276016. The negative root, −19.824228, also squares to 393.
Is the square root of 393 rational or irrational?
Irrational. 393 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 √393 be simplified?
No. 393 = 3 × 131 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √393 rounded to two decimal places?
√393 ≈ 19.82 to two decimal places (19.8 to one, 19.824 to three). Check: 19.82² = 392.8324, close to 393.