√350 at a glance
- Exact value
- 5√14
- Decimal (10 places)
- 18.7082869339
- Rounded
- 18.7 · 18.71 · 18.708
- Perfect square?
- No — between 18² and 19²
- Rational?
- Irrational
- Both square roots
- ±18.708287
- Prime factorization
- 2 × 5² × 7
- Cube root
- 7.047299
How to simplify √350
Look for the largest perfect square that divides 350. Here it is 25 (5²), because 350 = 25 × 14 and 14 has no square factor left:
The prime factorization tells the same story: 350 = 2 × 5² × 7. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 2 × 7 stays inside.
Check: (5√14)² = 5² × 14 = 25 × 14 = 350. As a decimal, 5√14 = 5 × 3.7416573868 ≈ 18.7082869339.
Where √350 sits between perfect squares
324 = 18² and 361 = 19² are the nearest perfect squares, so √350 lies between 18 and 19. 350 is 26 above 324 and 11 below 361, so the root is closer to 19.
- Straight line between 324 and 361: 18.7027 (0.03% low)
- Tangent from 18, i.e. 18 + 26 ÷ 36: 18.7222 (0.07% high)
- Tangent from 19, i.e. 19 − 11 ÷ 38: 18.7105 (0.01% high)
For √350 the tangent at 19 wins, missing by only 0.0022. Tangent estimates shine when the number sits close to a perfect square — here 350 is just 11 below 361.
Finding √350 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 | 350 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 18.4210526316 | 18.7105263158 | 2 |
| 2 | 18.7105263158 | 18.7060478200 | 18.7082870679 | 6 |
| 3 | 18.7082870679 | 18.7082867999 | 18.7082869339 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √350 = 18.7082869339 to every decimal shown.
√350 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √350 the pattern is [18; 1, 2, 2, 2, 1, 36] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √350 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 | 7.1 × 10⁻¹ |
| 19/1 | 19.0000000000 | 2.9 × 10⁻¹ |
| 56/3 | 18.6666666667 | 4.2 × 10⁻² |
| 131/7 | 18.7142857143 | 6.0 × 10⁻³ |
| 318/17 | 18.7058823529 | 2.4 × 10⁻³ |
| 449/24 | 18.7083333333 | 4.6 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 350y² = 1. Its smallest solution in positive whole numbers is x = 449, y = 24.
√350 in geometry and everyday measurements
- A square patio or deck of 350 square feet is about 18.71 ft (18 ft 8 in) on each side, so edging all the way around takes 4 × √350 ≈ 74.8 ft.
- 350 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √350 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 18 box, because 1² + 5² + 18² = 350.
- Since √350 = 5√14, a length of √350 is exactly 5 copies of the length √14 laid end to end.
Square roots near √350 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √347 | √347 | 18.6279 | No |
| √348 | 2√87 | 18.6548 | No |
| √349 | √349 | 18.6815 | No |
| √350 | 5√14 | 18.7083 | No |
| √351 | 3√39 | 18.7350 | No |
| √352 | 4√22 | 18.7617 | No |
| √353 | √353 | 18.7883 | No |
- The cube root of 350 is about 7.047299.
- Squaring undoes the root: (√350)² = 350, while 350² = 122,500 — the number whose square root is 350.
Frequently asked questions
What is the square root of 350?
The square root of 350 is 5√14 in simplest radical form, which is about 18.7082869339. The negative root, −18.708287, also squares to 350.
Is the square root of 350 rational or irrational?
Irrational. 350 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 √350 be simplified?
Yes. The largest perfect square dividing 350 is 25, so √350 = √25 × √14 = 5√14.
What is √350 rounded to two decimal places?
√350 ≈ 18.71 to two decimal places (18.7 to one, 18.708 to three). Check: 18.71² = 350.0641, close to 350.