√391 at a glance
- Exact value
- √391
- Decimal (10 places)
- 19.7737199333
- Rounded
- 19.8 · 19.77 · 19.774
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.773720
- Prime factorization
- 17 × 23
- Cube root
- 7.312383
How to simplify √391
The prime factorization of 391 is 17 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √391 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 391, 17 and 23 appear an odd number of times, so √391 is irrational and 19.7737199333 is a rounded value.
Where √391 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √391 lies between 19 and 20. 391 is 30 above 361 and 9 below 400, so the root is closer to 20.
- Straight line between 361 and 400: 19.7692 (0.02% low)
- Tangent from 19, i.e. 19 + 30 ÷ 38: 19.7895 (0.08% high)
- Tangent from 20, i.e. 20 − 9 ÷ 40: 19.7750 (0.01% high)
For √391 the tangent at 20 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 391 is just 9 below 400.
Finding √391 with the Babylonian method
If a guess is too big, 391 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√391) in one step.
Start from the nearest whole number, 20 (20² = 400):
| Step | Guess x | 391 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 19.5500000000 | 19.7750000000 | 2 |
| 2 | 19.7750000000 | 19.7724399494 | 19.7737199747 | 7 |
| 3 | 19.7737199747 | 19.7737198919 | 19.7737199333 | 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 √391 = 19.7737199333 to every decimal shown.
√391 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √391 the pattern is [19; 1, 3, 2, 2, 1, 1, 2, 19, 2, 1, 1, 2, …] 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 √391 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 | 7.7 × 10⁻¹ |
| 20/1 | 20.0000000000 | 2.3 × 10⁻¹ |
| 79/4 | 19.7500000000 | 2.4 × 10⁻² |
| 178/9 | 19.7777777778 | 4.1 × 10⁻³ |
| 435/22 | 19.7727272727 | 9.9 × 10⁻⁴ |
| 613/31 | 19.7741935484 | 4.7 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 391y² = 1. Its smallest solution in positive whole numbers is x = 7,338,680, y = 371,133.
√391 in geometry and everyday measurements
- A square patio or deck of 391 square feet is about 19.77 ft (19 ft 9 in) on each side, so edging all the way around takes 4 × √391 ≈ 79.1 ft.
- 391 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √391 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √391 as its space diagonal.
Square roots near √391 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √394 | √394 | 19.8494 | No |
- The cube root of 391 is about 7.312383.
- Squaring undoes the root: (√391)² = 391, while 391² = 152,881 — the number whose square root is 391.
Frequently asked questions
What is the square root of 391?
The square root of 391 is √391, about 19.7737199333. The negative root, −19.773720, also squares to 391.
Is the square root of 391 rational or irrational?
Irrational. 391 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 √391 be simplified?
No. 391 = 17 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √391 rounded to two decimal places?
√391 ≈ 19.77 to two decimal places (19.8 to one, 19.774 to three). Check: 19.77² = 390.8529, close to 391.