√14 at a glance
- Exact value
- √14
- Decimal (10 places)
- 3.7416573868
- Rounded
- 3.7 · 3.74 · 3.742
- Perfect square?
- No — between 3² and 4²
- Rational?
- Irrational
- Both square roots
- ±3.741657
- Prime factorization
- 2 × 7
- Cube root
- 2.410142
How to simplify √14
The prime factorization of 14 is 2 × 7. Every prime appears only once, so there is no pair to bring outside the radical — √14 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 14, 2 and 7 appear an odd number of times, so √14 is irrational and 3.7416573868 is a rounded value.
Where √14 sits between perfect squares
9 = 3² and 16 = 4² are the nearest perfect squares, so √14 lies between 3 and 4. 14 is 5 above 9 and 2 below 16, so the root is closer to 4.
- Straight line between 9 and 16: 3.7143 (0.73% low)
- Tangent from 3, i.e. 3 + 5 ÷ 6: 3.8333 (2.45% high)
- Tangent from 4, i.e. 4 − 2 ÷ 8: 3.7500 (0.22% high)
For √14 the tangent at 4 wins, missing by only 0.0083. Tangent estimates shine when the number sits close to a perfect square — here 14 is just 2 below 16.
Finding √14 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, 4 (4² = 16):
| Step | Guess x | 14 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 4.0000000000 | 3.5000000000 | 3.7500000000 | 2 |
| 2 | 3.7500000000 | 3.7333333333 | 3.7416666667 | 5 |
| 3 | 3.7416666667 | 3.7416481069 | 3.7416573868 | 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 √14 = 3.7416573868 to every decimal shown.
√14 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √14 the pattern is [3; 1, 2, 1, 6] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √14 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 |
|---|---|---|
| 3/1 | 3.0000000000 | 7.4 × 10⁻¹ |
| 4/1 | 4.0000000000 | 2.6 × 10⁻¹ |
| 11/3 | 3.6666666667 | 7.5 × 10⁻² |
| 15/4 | 3.7500000000 | 8.3 × 10⁻³ |
| 101/27 | 3.7407407407 | 9.2 × 10⁻⁴ |
| 116/31 | 3.7419354839 | 2.8 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 14y² = 1. Its smallest solution in positive whole numbers is x = 15, y = 4.
√14 in geometry and everyday measurements
- A square tile with an area of 14 square inches has sides about 3.742 in long.
- 14 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 √14 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 3 box, because 1² + 2² + 3² = 14.
Square roots near √14 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √11 | √11 | 3.3166 | No |
| √12 | 2√3 | 3.4641 | No |
| √13 | √13 | 3.6056 | No |
| √14 | √14 | 3.7417 | No |
| √15 | √15 | 3.8730 | No |
| √16 | 4 | 4.0000 | Yes |
| √17 | √17 | 4.1231 | No |
- The cube root of 14 is about 2.410142.
- Four times the radicand doubles the root: √56 = 2 × √14 ≈ 7.483315.
Frequently asked questions
What is the square root of 14?
The square root of 14 is √14, about 3.7416573868. The negative root, −3.741657, also squares to 14.
Is the square root of 14 rational or irrational?
Irrational. 14 is not a perfect square — it falls between 9 and 16 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √14 be simplified?
No. 14 = 2 × 7 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √14 rounded to two decimal places?
√14 ≈ 3.74 to two decimal places (3.7 to one, 3.742 to three). Check: 3.74² = 13.9876, close to 14.