√806 at a glance
- Exact value
- √806
- Decimal (10 places)
- 28.3901391332
- Rounded
- 28.4 · 28.39 · 28.390
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.390139
- Prime factorization
- 2 × 13 × 31
- Cube root
- 9.306328
How to simplify √806
The prime factorization of 806 is 2 × 13 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √806 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 806, 2, 13 and 31 appear an odd number of times, so √806 is irrational and 28.3901391332 is a rounded value.
Where √806 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √806 lies between 28 and 29. 806 is 22 above 784 and 35 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.3860 (0.01% low)
- Tangent from 28, i.e. 28 + 22 ÷ 56: 28.3929 (0.01% high)
- Tangent from 29, i.e. 29 − 35 ÷ 58: 28.3966 (0.02% high)
For √806 the tangent at 28 wins, missing by only 0.0027. Tangent estimates shine when the number sits close to a perfect square — here 806 is just 22 above 784.
Finding √806 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, 28 (28² = 784):
| Step | Guess x | 806 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.7857142857 | 28.3928571429 | 2 |
| 2 | 28.3928571429 | 28.3874213836 | 28.3901392633 | 6 |
| 3 | 28.3901392633 | 28.3901390031 | 28.3901391332 | 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 √806 = 28.3901391332 to every decimal shown.
√806 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √806 the pattern is [28; 2, 1, 1, 3, 2, 5, 4, 5, 2, 3, 1, 1, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √806 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 | 3.9 × 10⁻¹ |
| 57/2 | 28.5000000000 | 1.1 × 10⁻¹ |
| 85/3 | 28.3333333333 | 5.7 × 10⁻² |
| 142/5 | 28.4000000000 | 9.9 × 10⁻³ |
| 511/18 | 28.3888888889 | 1.3 × 10⁻³ |
| 1,164/41 | 28.3902439024 | 1.0 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 806y² = 1. Its smallest solution in positive whole numbers is x = 6,166,395, y = 217,202.
√806 in geometry and everyday measurements
- 806 square feet is 74.9 m². Laid out as a square — a small house footprint or a lot — it is about 28.39 ft (28 ft 5 in) on a side.
- 806 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √806 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 19 × 21 box, because 2² + 19² + 21² = 806.
Square roots near √806 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √803 | √803 | 28.3373 | No |
| √804 | 2√201 | 28.3549 | No |
| √805 | √805 | 28.3725 | No |
| √806 | √806 | 28.3901 | No |
| √807 | √807 | 28.4077 | No |
| √808 | 2√202 | 28.4253 | No |
| √809 | √809 | 28.4429 | No |
- The cube root of 806 is about 9.306328.
- Squaring undoes the root: (√806)² = 806, while 806² = 649,636 — the number whose square root is 806.
Frequently asked questions
What is the square root of 806?
The square root of 806 is √806, about 28.3901391332. The negative root, −28.390139, also squares to 806.
Is the square root of 806 rational or irrational?
Irrational. 806 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 √806 be simplified?
No. 806 = 2 × 13 × 31 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √806 rounded to two decimal places?
√806 ≈ 28.39 to two decimal places (28.4 to one, 28.390 to three). Check: 28.39² = 805.9921, close to 806.