√805 at a glance
- Exact value
- √805
- Decimal (10 places)
- 28.3725219182
- Rounded
- 28.4 · 28.37 · 28.373
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.372522
- Prime factorization
- 5 × 7 × 23
- Cube root
- 9.302477
How to simplify √805
The prime factorization of 805 is 5 × 7 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √805 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 805, 5, 7 and 23 appear an odd number of times, so √805 is irrational and 28.3725219182 is a rounded value.
Where √805 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √805 lies between 28 and 29. 805 is 21 above 784 and 36 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.3684 (0.01% low)
- Tangent from 28, i.e. 28 + 21 ÷ 56: 28.3750 (0.01% high)
- Tangent from 29, i.e. 29 − 36 ÷ 58: 28.3793 (0.02% high)
For √805 the tangent at 28 wins, missing by only 0.0025. Tangent estimates shine when the number sits close to a perfect square — here 805 is just 21 above 784.
Finding √805 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 805: following the tangent line down to zero simplifies to averaging x with 805 ÷ x.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 805 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.7500000000 | 28.3750000000 | 2 |
| 2 | 28.3750000000 | 28.3700440529 | 28.3725220264 | 6 |
| 3 | 28.3725220264 | 28.3725218100 | 28.3725219182 | 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 √805 = 28.3725219182 to every decimal shown.
√805 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √805 the pattern is [28; 2, 1, 2, 5, 1, 13, 2, 1, 10, 1, 2, 13, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √805 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 | 3.7 × 10⁻¹ |
| 57/2 | 28.5000000000 | 1.3 × 10⁻¹ |
| 85/3 | 28.3333333333 | 3.9 × 10⁻² |
| 227/8 | 28.3750000000 | 2.5 × 10⁻³ |
| 1,220/43 | 28.3720930233 | 4.3 × 10⁻⁴ |
| 1,447/51 | 28.3725490196 | 2.7 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 805y² = 1. Its smallest solution in positive whole numbers is x = 1,514,868,641, y = 53,392,104.
√805 in geometry and everyday measurements
- 805 square feet is 74.8 m². Laid out as a square — a small house footprint or a lot — it is about 28.37 ft (28 ft 4 in) on a side.
- 805 is not a sum of two whole-number squares — the prime factor 7 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √805 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 15 × 24 box, because 2² + 15² + 24² = 805.
Square roots near √805 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √802 | √802 | 28.3196 | No |
| √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 |
- The cube root of 805 is about 9.302477.
- Squaring undoes the root: (√805)² = 805, while 805² = 648,025 — the number whose square root is 805.
Frequently asked questions
What is the square root of 805?
The square root of 805 is √805, about 28.3725219182. The negative root, −28.372522, also squares to 805.
Is the square root of 805 rational or irrational?
Irrational. 805 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 √805 be simplified?
No. 805 = 5 × 7 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √805 rounded to two decimal places?
√805 ≈ 28.37 to two decimal places (28.4 to one, 28.373 to three). Check: 28.37² = 804.8569, close to 805.