√355 at a glance
- Exact value
- √355
- Decimal (10 places)
- 18.8414436814
- Rounded
- 18.8 · 18.84 · 18.841
- Perfect square?
- No — between 18² and 19²
- Rational?
- Irrational
- Both square roots
- ±18.841444
- Prime factorization
- 5 × 71
- Cube root
- 7.080699
How to simplify √355
The prime factorization of 355 is 5 × 71. Every prime appears only once, so there is no pair to bring outside the radical — √355 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 355, 5 and 71 appear an odd number of times, so √355 is irrational and 18.8414436814 is a rounded value.
Where √355 sits between perfect squares
324 = 18² and 361 = 19² are the nearest perfect squares, so √355 lies between 18 and 19. 355 is 31 above 324 and 6 below 361, so the root is closer to 19.
- Straight line between 324 and 361: 18.8378 (0.02% low)
- Tangent from 18, i.e. 18 + 31 ÷ 36: 18.8611 (0.1% high)
- Tangent from 19, i.e. 19 − 6 ÷ 38: 18.8421 (0% high)
For √355 the tangent at 19 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 355 is just 6 below 361.
Finding √355 with the Babylonian method
If a guess is too big, 355 ÷ 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√355) in one step.
Start from the nearest whole number, 19 (19² = 361):
| Step | Guess x | 355 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 18.6842105263 | 18.8421052632 | 3 |
| 2 | 18.8421052632 | 18.8407821229 | 18.8414436930 | 7 |
| 3 | 18.8414436930 | 18.8414436698 | 18.8414436814 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √355 = 18.8414436814 to every decimal shown.
√355 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √355 the pattern is [18; 1, 5, 3, 3, 1, 6, 1, 3, 3, 5, 1, 36] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √355 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 |
|---|---|---|
| 18/1 | 18.0000000000 | 8.4 × 10⁻¹ |
| 19/1 | 19.0000000000 | 1.6 × 10⁻¹ |
| 113/6 | 18.8333333333 | 8.1 × 10⁻³ |
| 358/19 | 18.8421052632 | 6.6 × 10⁻⁴ |
| 1,187/63 | 18.8412698413 | 1.7 × 10⁻⁴ |
| 1,545/82 | 18.8414634146 | 2.0 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 355y² = 1. Its smallest solution in positive whole numbers is x = 954,809, y = 50,676.
√355 in geometry and everyday measurements
- A square patio or deck of 355 square feet is about 18.84 ft (18 ft 10 in) on each side, so edging all the way around takes 4 × √355 ≈ 75.4 ft.
- 355 is not a sum of two whole-number squares — the prime factor 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √355 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 11 × 15 box, because 3² + 11² + 15² = 355.
Square roots near √355 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √352 | 4√22 | 18.7617 | No |
| √353 | √353 | 18.7883 | No |
| √354 | √354 | 18.8149 | No |
| √355 | √355 | 18.8414 | No |
| √356 | 2√89 | 18.8680 | No |
| √357 | √357 | 18.8944 | No |
| √358 | √358 | 18.9209 | No |
- The cube root of 355 is about 7.080699.
- Squaring undoes the root: (√355)² = 355, while 355² = 126,025 — the number whose square root is 355.
Frequently asked questions
What is the square root of 355?
The square root of 355 is √355, about 18.8414436814. The negative root, −18.841444, also squares to 355.
Is the square root of 355 rational or irrational?
Irrational. 355 is not a perfect square — it falls between 324 and 361 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √355 be simplified?
No. 355 = 5 × 71 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √355 rounded to two decimal places?
√355 ≈ 18.84 to two decimal places (18.8 to one, 18.841 to three). Check: 18.84² = 354.9456, close to 355.