√395 at a glance
- Exact value
- √395
- Decimal (10 places)
- 19.8746069144
- Rounded
- 19.9 · 19.87 · 19.875
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.874607
- Prime factorization
- 5 × 79
- Cube root
- 7.337234
How to simplify √395
The prime factorization of 395 is 5 × 79. Every prime appears only once, so there is no pair to bring outside the radical — √395 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 395, 5 and 79 appear an odd number of times, so √395 is irrational and 19.8746069144 is a rounded value.
Where √395 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √395 lies between 19 and 20. 395 is 34 above 361 and 5 below 400, so the root is closer to 20.
- Straight line between 361 and 400: 19.8718 (0.01% low)
- Tangent from 19, i.e. 19 + 34 ÷ 38: 19.8947 (0.1% high)
- Tangent from 20, i.e. 20 − 5 ÷ 40: 19.8750 (0% high)
For √395 the tangent at 20 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 395 is just 5 below 400.
Finding √395 with the Babylonian method
If a guess is too big, 395 ÷ 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√395) in one step.
Start from the nearest whole number, 20 (20² = 400):
| Step | Guess x | 395 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 19.7500000000 | 19.8750000000 | 3 |
| 2 | 19.8750000000 | 19.8742138365 | 19.8746069182 | 8 |
| 3 | 19.8746069182 | 19.8746069105 | 19.8746069144 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √395 = 19.8746069144 to every decimal shown.
√395 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √395 the pattern is [19; 1, 6, 1, 38] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √395 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 | 8.7 × 10⁻¹ |
| 20/1 | 20.0000000000 | 1.3 × 10⁻¹ |
| 139/7 | 19.8571428571 | 1.7 × 10⁻² |
| 159/8 | 19.8750000000 | 3.9 × 10⁻⁴ |
| 6,181/311 | 19.8745980707 | 8.8 × 10⁻⁶ |
| 6,340/319 | 19.8746081505 | 1.2 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 395y² = 1. Its smallest solution in positive whole numbers is x = 159, y = 8.
√395 in geometry and everyday measurements
- A square patio or deck of 395 square feet is about 19.87 ft (19 ft 10 in) on each side, so edging all the way around takes 4 × √395 ≈ 79.5 ft.
- 395 is not a sum of two whole-number squares — the prime factor 79 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √395 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 13 × 15 box, because 1² + 13² + 15² = 395.
Square roots near √395 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √397 | √397 | 19.9249 | No |
| √398 | √398 | 19.9499 | No |
- The cube root of 395 is about 7.337234.
- Squaring undoes the root: (√395)² = 395, while 395² = 156,025 — the number whose square root is 395.
Frequently asked questions
What is the square root of 395?
The square root of 395 is √395, about 19.8746069144. The negative root, −19.874607, also squares to 395.
Is the square root of 395 rational or irrational?
Irrational. 395 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 √395 be simplified?
No. 395 = 5 × 79 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √395 rounded to two decimal places?
√395 ≈ 19.87 to two decimal places (19.9 to one, 19.875 to three). Check: 19.87² = 394.8169, close to 395.