√814 at a glance
- Exact value
- √814
- Decimal (10 places)
- 28.5306852354
- Rounded
- 28.5 · 28.53 · 28.531
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.530685
- Prime factorization
- 2 × 11 × 37
- Cube root
- 9.337017
How to simplify √814
The prime factorization of 814 is 2 × 11 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √814 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 814, 2, 11 and 37 appear an odd number of times, so √814 is irrational and 28.5306852354 is a rounded value.
Where √814 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √814 lies between 28 and 29. 814 is 30 above 784 and 27 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.5263 (0.02% low)
- Tangent from 28, i.e. 28 + 30 ÷ 56: 28.5357 (0.02% high)
- Tangent from 29, i.e. 29 − 27 ÷ 58: 28.5345 (0.01% high)
For √814 the tangent at 29 wins, missing by only 0.0038. Tangent estimates shine when the number sits close to a perfect square — here 814 is just 27 below 841.
Finding √814 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, 29 (29² = 841):
| Step | Guess x | 814 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.0689655172 | 28.5344827586 | 2 |
| 2 | 28.5344827586 | 28.5268882175 | 28.5306854881 | 6 |
| 3 | 28.5306854881 | 28.5306849827 | 28.5306852354 | 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 √814 = 28.5306852354 to every decimal shown.
√814 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √814 the pattern is [28; 1, 1, 7, 1, 1, 1, 5, 18, 1, 5, 2, 1, …] with the block of 28 terms after the semicolon repeating forever (only the first 12 of the 28 are shown). A pattern that never ends is one more proof that √814 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 |
|---|---|---|
| 28/1 | 28.0000000000 | 5.3 × 10⁻¹ |
| 29/1 | 29.0000000000 | 4.7 × 10⁻¹ |
| 57/2 | 28.5000000000 | 3.1 × 10⁻² |
| 428/15 | 28.5333333333 | 2.6 × 10⁻³ |
| 485/17 | 28.5294117647 | 1.3 × 10⁻³ |
| 913/32 | 28.5312500000 | 5.6 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 814y² = 1. Its smallest solution in positive whole numbers is x = 4,206,992,174,549, y = 147,454,999,410 — 13 digits for x, even though 814 is small, which is what makes Pell’s equation famous.
√814 in geometry and everyday measurements
- 814 square feet is 75.6 m². Laid out as a square — a small house footprint or a lot — it is about 28.53 ft (28 ft 6 in) on a side.
- 814 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √814 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 9 × 27 box, because 2² + 9² + 27² = 814.
Square roots near √814 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √811 | √811 | 28.4781 | No |
| √812 | 2√203 | 28.4956 | No |
| √813 | √813 | 28.5132 | No |
| √814 | √814 | 28.5307 | No |
| √815 | √815 | 28.5482 | No |
| √816 | 4√51 | 28.5657 | No |
| √817 | √817 | 28.5832 | No |
- The cube root of 814 is about 9.337017.
- Squaring undoes the root: (√814)² = 814, while 814² = 662,596 — the number whose square root is 814.
Frequently asked questions
What is the square root of 814?
The square root of 814 is √814, about 28.5306852354. The negative root, −28.530685, also squares to 814.
Is the square root of 814 rational or irrational?
Irrational. 814 is not a perfect square — it falls between 784 and 841 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √814 be simplified?
No. 814 = 2 × 11 × 37 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √814 rounded to two decimal places?
√814 ≈ 28.53 to two decimal places (28.5 to one, 28.531 to three). Check: 28.53² = 813.9609, close to 814.