√398 at a glance
- Exact value
- √398
- Decimal (10 places)
- 19.9499373433
- Rounded
- 19.9 · 19.95 · 19.950
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.949937
- Prime factorization
- 2 × 199
- Cube root
- 7.355762
How to simplify √398
The prime factorization of 398 is 2 × 199. Every prime appears only once, so there is no pair to bring outside the radical — √398 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 398, 2 and 199 appear an odd number of times, so √398 is irrational and 19.9499373433 is a rounded value.
Where √398 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √398 lies between 19 and 20. 398 is 37 above 361 and 2 below 400, so the root is closer to 20.
- Straight line between 361 and 400: 19.9487 (0.01% low)
- Tangent from 19, i.e. 19 + 37 ÷ 38: 19.9737 (0.12% high)
- Tangent from 20, i.e. 20 − 2 ÷ 40: 19.9500 (0% high)
For √398 the tangent at 20 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 398 is just 2 below 400.
Finding √398 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 | 398 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 19.9000000000 | 19.9500000000 | 4 |
| 2 | 19.9500000000 | 19.9498746867 | 19.9499373434 | 10 |
| 3 | 19.9499373434 | 19.9499373432 | 19.9499373433 | all 10 shown |
The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √398 = 19.9499373433 to every decimal shown.
√398 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √398 the pattern is [19; 1, 18, 1, 38] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √398 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 | 9.5 × 10⁻¹ |
| 20/1 | 20.0000000000 | 5.0 × 10⁻² |
| 379/19 | 19.9473684211 | 2.6 × 10⁻³ |
| 399/20 | 19.9500000000 | 6.3 × 10⁻⁵ |
| 15,541/779 | 19.9499358151 | 1.5 × 10⁻⁶ |
| 15,940/799 | 19.9499374218 | 7.9 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 398y² = 1. Its smallest solution in positive whole numbers is x = 399, y = 20.
√398 in geometry and everyday measurements
- A square patio or deck of 398 square feet is about 19.95 ft (19 ft 11 in) on each side, so edging all the way around takes 4 × √398 ≈ 79.8 ft.
- 398 is not a sum of two whole-number squares — the prime factor 199 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √398 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 19 box, because 1² + 6² + 19² = 398.
Square roots near √398 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √395 | √395 | 19.8746 | No |
| √396 | 6√11 | 19.8997 | No |
| √397 | √397 | 19.9249 | No |
| √398 | √398 | 19.9499 | No |
| √399 | √399 | 19.9750 | No |
| √400 | 20 | 20.0000 | Yes |
| √401 | √401 | 20.0250 | No |
- The cube root of 398 is about 7.355762.
- Squaring undoes the root: (√398)² = 398, while 398² = 158,404 — the number whose square root is 398.
Frequently asked questions
What is the square root of 398?
The square root of 398 is √398, about 19.9499373433. The negative root, −19.949937, also squares to 398.
Is the square root of 398 rational or irrational?
Irrational. 398 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 √398 be simplified?
No. 398 = 2 × 199 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √398 rounded to two decimal places?
√398 ≈ 19.95 to two decimal places (19.9 to one, 19.950 to three). Check: 19.95² = 398.0025, close to 398.