√342 at a glance
- Exact value
- 3√38
- Decimal (10 places)
- 18.4932420089
- Rounded
- 18.5 · 18.49 · 18.493
- Perfect square?
- No — between 18² and 19²
- Rational?
- Irrational
- Both square roots
- ±18.493242
- Prime factorization
- 2 × 3² × 19
- Cube root
- 6.993191
How to simplify √342
Look for the largest perfect square that divides 342. Here it is 9 (3²), because 342 = 9 × 38 and 38 has no square factor left:
The prime factorization tells the same story: 342 = 2 × 3² × 19. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 19 stays inside.
Check: (3√38)² = 3² × 38 = 9 × 38 = 342. As a decimal, 3√38 = 3 × 6.164414003 ≈ 18.4932420089.
Where √342 sits between perfect squares
324 = 18² and 361 = 19² are the nearest perfect squares, so √342 lies between 18 and 19. 342 is 18 above 324 and 19 below 361, so the root is closer to 18.
- Straight line between 324 and 361: 18.4865 (0.04% low)
- Tangent from 18, i.e. 18 + 18 ÷ 36: 18.5000 (0.04% high)
- Tangent from 19, i.e. 19 − 19 ÷ 38: 18.5000 (0.04% high)
For √342 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.
Finding √342 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, 18 (18² = 324):
| Step | Guess x | 342 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 18.0000000000 | 19.0000000000 | 18.5000000000 | 2 |
| 2 | 18.5000000000 | 18.4864864865 | 18.4932432432 | 5 |
| 3 | 18.4932432432 | 18.4932407746 | 18.4932420089 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √342 = 18.4932420089 to every decimal shown.
√342 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √342 the pattern is [18; 2, 36] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √342 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 | 4.9 × 10⁻¹ |
| 37/2 | 18.5000000000 | 6.8 × 10⁻³ |
| 1,350/73 | 18.4931506849 | 9.1 × 10⁻⁵ |
| 2,737/148 | 18.4932432432 | 1.2 × 10⁻⁶ |
| 99,882/5,401 | 18.4932419922 | 1.7 × 10⁻⁸ |
| 202,501/10,950 | 18.4932420091 | 2.3 × 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 342y² = 1. Its smallest solution in positive whole numbers is x = 37, y = 2.
√342 in geometry and everyday measurements
- A square patio or deck of 342 square feet is about 18.49 ft (18 ft 6 in) on each side, so edging all the way around takes 4 × √342 ≈ 74 ft.
- 342 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √342 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 7 × 17 box, because 2² + 7² + 17² = 342.
- Since √342 = 3√38, a length of √342 is exactly 3 copies of the length √38 laid end to end.
Square roots near √342 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √339 | √339 | 18.4120 | No |
| √340 | 2√85 | 18.4391 | No |
| √341 | √341 | 18.4662 | No |
| √342 | 3√38 | 18.4932 | No |
| √343 | 7√7 | 18.5203 | No |
| √344 | 2√86 | 18.5472 | No |
| √345 | √345 | 18.5742 | No |
- The cube root of 342 is about 6.993191.
- Squaring undoes the root: (√342)² = 342, while 342² = 116,964 — the number whose square root is 342.
Frequently asked questions
What is the square root of 342?
The square root of 342 is 3√38 in simplest radical form, which is about 18.4932420089. The negative root, −18.493242, also squares to 342.
Is the square root of 342 rational or irrational?
Irrational. 342 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 √342 be simplified?
Yes. The largest perfect square dividing 342 is 9, so √342 = √9 × √38 = 3√38.
What is √342 rounded to two decimal places?
√342 ≈ 18.49 to two decimal places (18.5 to one, 18.493 to three). Check: 18.49² = 341.8801, close to 342.