√390 at a glance
- Exact value
- √390
- Decimal (10 places)
- 19.7484176581
- Rounded
- 19.7 · 19.75 · 19.748
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.748418
- Prime factorization
- 2 × 3 × 5 × 13
- Cube root
- 7.306144
How to simplify √390
The prime factorization of 390 is 2 × 3 × 5 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √390 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 390, 2, 3, 5 and 13 appear an odd number of times, so √390 is irrational and 19.7484176581 is a rounded value.
Where √390 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √390 lies between 19 and 20. 390 is 29 above 361 and 10 below 400, so the root is closer to 20.
- Straight line between 361 and 400: 19.7436 (0.02% low)
- Tangent from 19, i.e. 19 + 29 ÷ 38: 19.7632 (0.07% high)
- Tangent from 20, i.e. 20 − 10 ÷ 40: 19.7500 (0.01% high)
For √390 the tangent at 20 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 390 is just 10 below 400.
Finding √390 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 | 390 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 19.5000000000 | 19.7500000000 | 2 |
| 2 | 19.7500000000 | 19.7468354430 | 19.7484177215 | 7 |
| 3 | 19.7484177215 | 19.7484175947 | 19.7484176581 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √390 = 19.7484176581 to every decimal shown.
√390 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √390 the pattern is [19; 1, 2, 1, 38] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √390 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 | 7.5 × 10⁻¹ |
| 20/1 | 20.0000000000 | 2.5 × 10⁻¹ |
| 59/3 | 19.6666666667 | 8.2 × 10⁻² |
| 79/4 | 19.7500000000 | 1.6 × 10⁻³ |
| 3,061/155 | 19.7483870968 | 3.1 × 10⁻⁵ |
| 3,140/159 | 19.7484276730 | 1.0 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 390y² = 1. Its smallest solution in positive whole numbers is x = 79, y = 4.
√390 in geometry and everyday measurements
- A square patio or deck of 390 square feet is about 19.75 ft (19 ft 9 in) on each side, so edging all the way around takes 4 × √390 ≈ 79 ft.
- 390 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √390 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 10 × 17 box, because 1² + 10² + 17² = 390.
Square roots near √390 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √387 | 3√43 | 19.6723 | No |
| √388 | 2√97 | 19.6977 | No |
| √389 | √389 | 19.7231 | No |
| √390 | √390 | 19.7484 | No |
| √391 | √391 | 19.7737 | No |
| √392 | 14√2 | 19.7990 | No |
| √393 | √393 | 19.8242 | No |
- The cube root of 390 is about 7.306144.
- Squaring undoes the root: (√390)² = 390, while 390² = 152,100 — the number whose square root is 390.
Frequently asked questions
What is the square root of 390?
The square root of 390 is √390, about 19.7484176581. The negative root, −19.748418, also squares to 390.
Is the square root of 390 rational or irrational?
Irrational. 390 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 √390 be simplified?
No. 390 = 2 × 3 × 5 × 13 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √390 rounded to two decimal places?
√390 ≈ 19.75 to two decimal places (19.7 to one, 19.748 to three). Check: 19.75² = 390.0625, close to 390.