√357 at a glance
- Exact value
- √357
- Decimal (10 places)
- 18.8944436277
- Rounded
- 18.9 · 18.89 · 18.894
- Perfect square?
- No — between 18² and 19²
- Rational?
- Irrational
- Both square roots
- ±18.894444
- Prime factorization
- 3 × 7 × 17
- Cube root
- 7.093971
How to simplify √357
The prime factorization of 357 is 3 × 7 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √357 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 357, 3, 7 and 17 appear an odd number of times, so √357 is irrational and 18.8944436277 is a rounded value.
Where √357 sits between perfect squares
324 = 18² and 361 = 19² are the nearest perfect squares, so √357 lies between 18 and 19. 357 is 33 above 324 and 4 below 361, so the root is closer to 19.
- Straight line between 324 and 361: 18.8919 (0.01% low)
- Tangent from 18, i.e. 18 + 33 ÷ 36: 18.9167 (0.12% high)
- Tangent from 19, i.e. 19 − 4 ÷ 38: 18.8947 (0% high)
For √357 the tangent at 19 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 357 is just 4 below 361.
Finding √357 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 357: following the tangent line down to zero simplifies to averaging x with 357 ÷ x.
Start from the nearest whole number, 19 (19² = 361):
| Step | Guess x | 357 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 18.7894736842 | 18.8947368421 | 3 |
| 2 | 18.8947368421 | 18.8941504178 | 18.8944436300 | 8 |
| 3 | 18.8944436300 | 18.8944436254 | 18.8944436277 | 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 √357 = 18.8944436277 to every decimal shown.
√357 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √357 the pattern is [18; 1, 8, 2, 8, 1, 36] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √357 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 | 8.9 × 10⁻¹ |
| 19/1 | 19.0000000000 | 1.1 × 10⁻¹ |
| 170/9 | 18.8888888889 | 5.6 × 10⁻³ |
| 359/19 | 18.8947368421 | 2.9 × 10⁻⁴ |
| 3,042/161 | 18.8944099379 | 3.4 × 10⁻⁵ |
| 3,401/180 | 18.8944444444 | 8.2 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 357y² = 1. Its smallest solution in positive whole numbers is x = 3,401, y = 180.
√357 in geometry and everyday measurements
- A square patio or deck of 357 square feet is about 18.89 ft (18 ft 11 in) on each side, so edging all the way around takes 4 × √357 ≈ 75.6 ft.
- 357 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √357 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 10 × 16 box, because 1² + 10² + 16² = 357.
Square roots near √357 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √359 | √359 | 18.9473 | No |
| √360 | 6√10 | 18.9737 | No |
- The cube root of 357 is about 7.093971.
- Squaring undoes the root: (√357)² = 357, while 357² = 127,449 — the number whose square root is 357.
Frequently asked questions
What is the square root of 357?
The square root of 357 is √357, about 18.8944436277. The negative root, −18.894444, also squares to 357.
Is the square root of 357 rational or irrational?
Irrational. 357 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 √357 be simplified?
No. 357 = 3 × 7 × 17 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √357 rounded to two decimal places?
√357 ≈ 18.89 to two decimal places (18.9 to one, 18.894 to three). Check: 18.89² = 356.8321, close to 357.