√214 at a glance
- Exact value
- √214
- Decimal (10 places)
- 14.6287388383
- Rounded
- 14.6 · 14.63 · 14.629
- Perfect square?
- No — between 14² and 15²
- Rational?
- Irrational
- Both square roots
- ±14.628739
- Prime factorization
- 2 × 107
- Cube root
- 5.981424
How to simplify √214
The prime factorization of 214 is 2 × 107. Every prime appears only once, so there is no pair to bring outside the radical — √214 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 214, 2 and 107 appear an odd number of times, so √214 is irrational and 14.6287388383 is a rounded value.
Where √214 sits between perfect squares
196 = 14² and 225 = 15² are the nearest perfect squares, so √214 lies between 14 and 15. 214 is 18 above 196 and 11 below 225, so the root is closer to 15.
- Straight line between 196 and 225: 14.6207 (0.06% low)
- Tangent from 14, i.e. 14 + 18 ÷ 28: 14.6429 (0.1% high)
- Tangent from 15, i.e. 15 − 11 ÷ 30: 14.6333 (0.03% high)
For √214 the tangent at 15 wins, missing by only 0.0046. Tangent estimates shine when the number sits close to a perfect square — here 214 is just 11 below 225.
Finding √214 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, 15 (15² = 225):
| Step | Guess x | 214 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 15.0000000000 | 14.2666666667 | 14.6333333333 | 2 |
| 2 | 14.6333333333 | 14.6241457859 | 14.6287395596 | 6 |
| 3 | 14.6287395596 | 14.6287381171 | 14.6287388383 | 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 √214 = 14.6287388383 to every decimal shown.
√214 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √214 the pattern is [14; 1, 1, 1, 2, 3, 1, 4, 9, 1, 1, 5, 3, …] 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 √214 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 |
|---|---|---|
| 14/1 | 14.0000000000 | 6.3 × 10⁻¹ |
| 15/1 | 15.0000000000 | 3.7 × 10⁻¹ |
| 29/2 | 14.5000000000 | 1.3 × 10⁻¹ |
| 44/3 | 14.6666666667 | 3.8 × 10⁻² |
| 117/8 | 14.6250000000 | 3.7 × 10⁻³ |
| 395/27 | 14.6296296296 | 8.9 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 214y² = 1. Its smallest solution in positive whole numbers is x = 695,359,189,925, y = 47,533,775,646.
√214 in geometry and everyday measurements
- A square patio or deck of 214 square feet is about 14.63 ft (14 ft 8 in) on each side, so edging all the way around takes 4 × √214 ≈ 58.5 ft.
- 214 is not a sum of two whole-number squares — the prime factor 107 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √214 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 3 × 14 box, because 3² + 3² + 14² = 214.
Square roots near √214 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √211 | √211 | 14.5258 | No |
| √212 | 2√53 | 14.5602 | No |
| √213 | √213 | 14.5945 | No |
| √214 | √214 | 14.6287 | No |
| √215 | √215 | 14.6629 | No |
| √216 | 6√6 | 14.6969 | No |
| √217 | √217 | 14.7309 | No |
- The cube root of 214 is about 5.981424.
- Four times the radicand doubles the root: √856 = 2 × √214 ≈ 29.257478.
Frequently asked questions
What is the square root of 214?
The square root of 214 is √214, about 14.6287388383. The negative root, −14.628739, also squares to 214.
Is the square root of 214 rational or irrational?
Irrational. 214 is not a perfect square — it falls between 196 and 225 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √214 be simplified?
No. 214 = 2 × 107 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √214 rounded to two decimal places?
√214 ≈ 14.63 to two decimal places (14.6 to one, 14.629 to three). Check: 14.63² = 214.0369, close to 214.