√191 at a glance
- Exact value
- √191
- Decimal (10 places)
- 13.8202749611
- Rounded
- 13.8 · 13.82 · 13.820
- Perfect square?
- No — between 13² and 14²
- Rational?
- Irrational
- Both square roots
- ±13.820275
- Prime factorization
- 191
- Cube root
- 5.758965
How to simplify √191
191 is a prime number, so its only factors are 1 and 191. There is no perfect-square factor to pull out, which means √191 is already in its simplest radical form.
The square root of any prime is irrational. If √191 were a fraction a/b in lowest terms, then a² = 191b², so 191 would divide a — and then 191 would divide b too, contradicting “lowest terms.” That is why the decimal 13.8202749611 is only a rounded value.
Where √191 sits between perfect squares
169 = 13² and 196 = 14² are the nearest perfect squares, so √191 lies between 13 and 14. 191 is 22 above 169 and 5 below 196, so the root is closer to 14.
- Straight line between 169 and 196: 13.8148 (0.04% low)
- Tangent from 13, i.e. 13 + 22 ÷ 26: 13.8462 (0.19% high)
- Tangent from 14, i.e. 14 − 5 ÷ 28: 13.8214 (0.01% high)
For √191 the tangent at 14 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 191 is just 5 below 196.
Finding √191 with the Babylonian method
If a guess is too big, 191 ÷ 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√191) in one step.
Start from the nearest whole number, 14 (14² = 196):
| Step | Guess x | 191 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 14.0000000000 | 13.6428571429 | 13.8214285714 | 2 |
| 2 | 13.8214285714 | 13.8191214470 | 13.8202750092 | 7 |
| 3 | 13.8202750092 | 13.8202749129 | 13.8202749611 | 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 √191 = 13.8202749611 to every decimal shown.
√191 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √191 the pattern is [13; 1, 4, 1, 1, 3, 2, 2, 13, 2, 2, 3, 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 √191 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 |
|---|---|---|
| 13/1 | 13.0000000000 | 8.2 × 10⁻¹ |
| 14/1 | 14.0000000000 | 1.8 × 10⁻¹ |
| 69/5 | 13.8000000000 | 2.0 × 10⁻² |
| 83/6 | 13.8333333333 | 1.3 × 10⁻² |
| 152/11 | 13.8181818182 | 2.1 × 10⁻³ |
| 539/39 | 13.8205128205 | 2.4 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 191y² = 1. Its smallest solution in positive whole numbers is x = 8,994,000, y = 650,783.
√191 in geometry and everyday measurements
- A square room or garden bed covering 191 square feet measures about 13.82 ft (13 ft 10 in) along each wall.
- 191 is not a sum of two whole-number squares — 191 is itself a prime that is one less than a multiple of 4, which rules that out — so √191 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 √191 as its space diagonal.
Square roots near √191 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √188 | 2√47 | 13.7113 | No |
| √189 | 3√21 | 13.7477 | No |
| √190 | √190 | 13.7840 | No |
| √191 | √191 | 13.8203 | No |
| √192 | 8√3 | 13.8564 | No |
| √193 | √193 | 13.8924 | No |
| √194 | √194 | 13.9284 | No |
- The cube root of 191 is about 5.758965.
- Four times the radicand doubles the root: √764 = 2 × √191 ≈ 27.64055.
Frequently asked questions
What is the square root of 191?
The square root of 191 is √191, about 13.8202749611. The negative root, −13.820275, also squares to 191.
Is the square root of 191 rational or irrational?
Irrational. 191 is not a perfect square — it falls between 169 and 196 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √191 be simplified?
No. 191 is prime, so there is no perfect square to take out of the radical.
What is √191 rounded to two decimal places?
√191 ≈ 13.82 to two decimal places (13.8 to one, 13.820 to three). Check: 13.82² = 190.9924, close to 191.