√371 at a glance
- Exact value
- √371
- Decimal (10 places)
- 19.2613602843
- Rounded
- 19.3 · 19.26 · 19.261
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.261360
- Prime factorization
- 7 × 53
- Cube root
- 7.185516
How to simplify √371
The prime factorization of 371 is 7 × 53. Every prime appears only once, so there is no pair to bring outside the radical — √371 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 371, 7 and 53 appear an odd number of times, so √371 is irrational and 19.2613602843 is a rounded value.
Where √371 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √371 lies between 19 and 20. 371 is 10 above 361 and 29 below 400, so the root is closer to 19.
- Straight line between 361 and 400: 19.2564 (0.03% low)
- Tangent from 19, i.e. 19 + 10 ÷ 38: 19.2632 (0.01% high)
- Tangent from 20, i.e. 20 − 29 ÷ 40: 19.2750 (0.07% high)
For √371 the tangent at 19 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 371 is just 10 above 361.
Finding √371 with the Babylonian method
If a guess is too big, 371 ÷ 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√371) in one step.
Start from the nearest whole number, 19 (19² = 361):
| Step | Guess x | 371 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 19.5263157895 | 19.2631578947 | 2 |
| 2 | 19.2631578947 | 19.2595628415 | 19.2613603681 | 7 |
| 3 | 19.2613603681 | 19.2613602004 | 19.2613602843 | 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 √371 = 19.2613602843 to every decimal shown.
√371 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √371 the pattern is [19; 3, 1, 4, 1, 3, 38] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √371 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 | 2.6 × 10⁻¹ |
| 58/3 | 19.3333333333 | 7.2 × 10⁻² |
| 77/4 | 19.2500000000 | 1.1 × 10⁻² |
| 366/19 | 19.2631578947 | 1.8 × 10⁻³ |
| 443/23 | 19.2608695652 | 4.9 × 10⁻⁴ |
| 1,695/88 | 19.2613636364 | 3.4 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 371y² = 1. Its smallest solution in positive whole numbers is x = 1,695, y = 88.
√371 in geometry and everyday measurements
- A square patio or deck of 371 square feet is about 19.26 ft (19 ft 3 in) on each side, so edging all the way around takes 4 × √371 ≈ 77 ft.
- 371 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √371 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 19 box, because 1² + 3² + 19² = 371.
Square roots near √371 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √368 | 4√23 | 19.1833 | No |
| √369 | 3√41 | 19.2094 | No |
| √370 | √370 | 19.2354 | No |
| √371 | √371 | 19.2614 | No |
| √372 | 2√93 | 19.2873 | No |
| √373 | √373 | 19.3132 | No |
| √374 | √374 | 19.3391 | No |
- The cube root of 371 is about 7.185516.
- Squaring undoes the root: (√371)² = 371, while 371² = 137,641 — the number whose square root is 371.
Frequently asked questions
What is the square root of 371?
The square root of 371 is √371, about 19.2613602843. The negative root, −19.261360, also squares to 371.
Is the square root of 371 rational or irrational?
Irrational. 371 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 √371 be simplified?
No. 371 = 7 × 53 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √371 rounded to two decimal places?
√371 ≈ 19.26 to two decimal places (19.3 to one, 19.261 to three). Check: 19.26² = 370.9476, close to 371.