√807 at a glance
- Exact value
- √807
- Decimal (10 places)
- 28.4077454227
- Rounded
- 28.4 · 28.41 · 28.408
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.407745
- Prime factorization
- 3 × 269
- Cube root
- 9.310175
How to simplify √807
The prime factorization of 807 is 3 × 269. Every prime appears only once, so there is no pair to bring outside the radical — √807 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 807, 3 and 269 appear an odd number of times, so √807 is irrational and 28.4077454227 is a rounded value.
Where √807 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √807 lies between 28 and 29. 807 is 23 above 784 and 34 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.4035 (0.01% low)
- Tangent from 28, i.e. 28 + 23 ÷ 56: 28.4107 (0.01% high)
- Tangent from 29, i.e. 29 − 34 ÷ 58: 28.4138 (0.02% high)
For √807 the tangent at 28 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 807 is just 23 above 784.
Finding √807 with the Babylonian method
If a guess is too big, 807 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√807) in one step.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 807 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.8214285714 | 28.4107142857 | 2 |
| 2 | 28.4107142857 | 28.4047768699 | 28.4077455778 | 6 |
| 3 | 28.4077455778 | 28.4077452676 | 28.4077454227 | 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 √807 = 28.4077454227 to every decimal shown.
√807 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √807 the pattern is [28; 2, 2, 4, 1, 3, 4, 9, 4, 3, 1, 4, 2, …] 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 √807 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 | 4.1 × 10⁻¹ |
| 57/2 | 28.5000000000 | 9.2 × 10⁻² |
| 142/5 | 28.4000000000 | 7.7 × 10⁻³ |
| 625/22 | 28.4090909091 | 1.3 × 10⁻³ |
| 767/27 | 28.4074074074 | 3.4 × 10⁻⁴ |
| 2,926/103 | 28.4077669903 | 2.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 807y² = 1. Its smallest solution in positive whole numbers is x = 51,841,948, y = 1,824,923.
√807 in geometry and everyday measurements
- 807 square feet is 75 m². Laid out as a square — a small house footprint or a lot — it is about 28.41 ft (28 ft 5 in) on a side.
- 807 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √807 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √807 as its space diagonal.
Square roots near √807 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √810 | 9√10 | 28.4605 | No |
- The cube root of 807 is about 9.310175.
- Squaring undoes the root: (√807)² = 807, while 807² = 651,249 — the number whose square root is 807.
Frequently asked questions
What is the square root of 807?
The square root of 807 is √807, about 28.4077454227. The negative root, −28.407745, also squares to 807.
Is the square root of 807 rational or irrational?
Irrational. 807 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 √807 be simplified?
No. 807 = 3 × 269 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √807 rounded to two decimal places?
√807 ≈ 28.41 to two decimal places (28.4 to one, 28.408 to three). Check: 28.41² = 807.1281, close to 807.