√358 at a glance
- Exact value
- √358
- Decimal (10 places)
- 18.9208879284
- Rounded
- 18.9 · 18.92 · 18.921
- Perfect square?
- No — between 18² and 19²
- Rational?
- Irrational
- Both square roots
- ±18.920888
- Prime factorization
- 2 × 179
- Cube root
- 7.100588
How to simplify √358
The prime factorization of 358 is 2 × 179. Every prime appears only once, so there is no pair to bring outside the radical — √358 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 358, 2 and 179 appear an odd number of times, so √358 is irrational and 18.9208879284 is a rounded value.
Where √358 sits between perfect squares
324 = 18² and 361 = 19² are the nearest perfect squares, so √358 lies between 18 and 19. 358 is 34 above 324 and 3 below 361, so the root is closer to 19.
- Straight line between 324 and 361: 18.9189 (0.01% low)
- Tangent from 18, i.e. 18 + 34 ÷ 36: 18.9444 (0.12% high)
- Tangent from 19, i.e. 19 − 3 ÷ 38: 18.9211 (0% high)
For √358 the tangent at 19 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 358 is just 3 below 361.
Finding √358 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, 19 (19² = 361):
| Step | Guess x | 358 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 18.8421052632 | 18.9210526316 | 3 |
| 2 | 18.9210526316 | 18.9207232267 | 18.9208879291 | 9 |
| 3 | 18.9208879291 | 18.9208879277 | 18.9208879284 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √358 = 18.9208879284 to every decimal shown.
√358 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √358 the pattern is [18; 1, 11, 1, 1, 1, 3, 1, 1, 4, 1, 5, 2, …] with the block of 26 terms after the semicolon repeating forever (only the first 12 of the 26 are shown). A pattern that never ends is one more proof that √358 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 | 9.2 × 10⁻¹ |
| 19/1 | 19.0000000000 | 7.9 × 10⁻² |
| 227/12 | 18.9166666667 | 4.2 × 10⁻³ |
| 246/13 | 18.9230769231 | 2.2 × 10⁻³ |
| 473/25 | 18.9200000000 | 8.9 × 10⁻⁴ |
| 719/38 | 18.9210526316 | 1.6 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 358y² = 1. Its smallest solution in positive whole numbers is x = 176,579,805,797, y = 9,332,532,726.
√358 in geometry and everyday measurements
- A square patio or deck of 358 square feet is about 18.92 ft (18 ft 11 in) on each side, so edging all the way around takes 4 × √358 ≈ 75.7 ft.
- 358 is not a sum of two whole-number squares — the prime factor 179 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √358 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 5 × 18 box, because 3² + 5² + 18² = 358.
Square roots near √358 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √361 | 19 | 19.0000 | Yes |
- The cube root of 358 is about 7.100588.
- Squaring undoes the root: (√358)² = 358, while 358² = 128,164 — the number whose square root is 358.
Frequently asked questions
What is the square root of 358?
The square root of 358 is √358, about 18.9208879284. The negative root, −18.920888, also squares to 358.
Is the square root of 358 rational or irrational?
Irrational. 358 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 √358 be simplified?
No. 358 = 2 × 179 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √358 rounded to two decimal places?
√358 ≈ 18.92 to two decimal places (18.9 to one, 18.921 to three). Check: 18.92² = 357.9664, close to 358.