√399 at a glance
- Exact value
- √399
- Decimal (10 places)
- 19.9749843554
- Rounded
- 20.0 · 19.97 · 19.975
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.974984
- Prime factorization
- 3 × 7 × 19
- Cube root
- 7.361918
How to simplify √399
The prime factorization of 399 is 3 × 7 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √399 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 399, 3, 7 and 19 appear an odd number of times, so √399 is irrational and 19.9749843554 is a rounded value.
Where √399 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √399 lies between 19 and 20. 399 is 38 above 361 and 1 below 400, so the root is closer to 20.
- Straight line between 361 and 400: 19.9744 (0% low)
- Tangent from 19, i.e. 19 + 38 ÷ 38: 20.0000 (0.13% high)
- Tangent from 20, i.e. 20 − 1 ÷ 40: 19.9750 (0% high)
For √399 the tangent at 20 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 399 is just 1 below 400.
Finding √399 with the Babylonian method
If a guess is too big, 399 ÷ 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√399) in one step.
Start from the nearest whole number, 20 (20² = 400):
| Step | Guess x | 399 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 19.9500000000 | 19.9750000000 | 4 |
| 2 | 19.9750000000 | 19.9749687109 | 19.9749843554 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √399 = 19.9749843554 to every decimal shown.
√399 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √399 the pattern is [19; 1, 38] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √399 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 | 9.7 × 10⁻¹ |
| 20/1 | 20.0000000000 | 2.5 × 10⁻² |
| 779/39 | 19.9743589744 | 6.3 × 10⁻⁴ |
| 799/40 | 19.9750000000 | 1.6 × 10⁻⁵ |
| 31,141/1,559 | 19.9749839641 | 3.9 × 10⁻⁷ |
| 31,940/1,599 | 19.9749843652 | 9.8 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 399y² = 1. Its smallest solution in positive whole numbers is x = 20, y = 1.
√399 in geometry and everyday measurements
- A square patio or deck of 399 square feet is about 19.97 ft (20 ft) on each side, so edging all the way around takes 4 × √399 ≈ 79.9 ft.
- 399 is not a sum of two whole-number squares — the prime factor 3, 7 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √399 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 √399 as its space diagonal.
Square roots near √399 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √402 | √402 | 20.0499 | No |
- The cube root of 399 is about 7.361918.
- Squaring undoes the root: (√399)² = 399, while 399² = 159,201 — the number whose square root is 399.
Frequently asked questions
What is the square root of 399?
The square root of 399 is √399, about 19.9749843554. The negative root, −19.974984, also squares to 399.
Is the square root of 399 rational or irrational?
Irrational. 399 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 √399 be simplified?
No. 399 = 3 × 7 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √399 rounded to two decimal places?
√399 ≈ 19.97 to two decimal places (20.0 to one, 19.975 to three). Check: 19.97² = 398.8009, close to 399.