√794 at a glance
- Exact value
- √794
- Decimal (10 places)
- 28.1780056072
- Rounded
- 28.2 · 28.18 · 28.178
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.178006
- Prime factorization
- 2 × 397
- Cube root
- 9.259911
How to simplify √794
The prime factorization of 794 is 2 × 397. Every prime appears only once, so there is no pair to bring outside the radical — √794 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 794, 2 and 397 appear an odd number of times, so √794 is irrational and 28.1780056072 is a rounded value.
Where √794 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √794 lies between 28 and 29. 794 is 10 above 784 and 47 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.1754 (0.01% low)
- Tangent from 28, i.e. 28 + 10 ÷ 56: 28.1786 (0% high)
- Tangent from 29, i.e. 29 − 47 ÷ 58: 28.1897 (0.04% high)
For √794 the tangent at 28 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 794 is just 10 above 784.
Finding √794 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 | 794 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.3571428571 | 28.1785714286 | 3 |
| 2 | 28.1785714286 | 28.1774397972 | 28.1780056129 | 8 |
| 3 | 28.1780056129 | 28.1780056015 | 28.1780056072 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √794 = 28.1780056072 to every decimal shown.
√794 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √794 the pattern is [28; 5, 1, 1, 1, 1, 1, 1, 1, 1, 5, 56] with the block of 11 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √794 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 | 1.8 × 10⁻¹ |
| 141/5 | 28.2000000000 | 2.2 × 10⁻² |
| 169/6 | 28.1666666667 | 1.1 × 10⁻² |
| 310/11 | 28.1818181818 | 3.8 × 10⁻³ |
| 479/17 | 28.1764705882 | 1.5 × 10⁻³ |
| 789/28 | 28.1785714286 | 5.7 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 794y² = 1. Its smallest solution in positive whole numbers is x = 1,828,310,451, y = 64,884,310. Because the period is odd, the equation with −1 on the right also has a solution: 30,235² − 794 × 1,073² = −1.
√794 in geometry and everyday measurements
- 794 square feet is 73.8 m². Laid out as a square — a small house footprint or a lot — it is about 28.18 ft (28 ft 2 in) on a side.
- 794 = 13² + 25², so by the Pythagorean theorem √794 is the diagonal of a 13 × 25 rectangle — and the distance between the points (0, 0) and (13, 25) on a grid.
Square roots near √794 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √791 | √791 | 28.1247 | No |
| √792 | 6√22 | 28.1425 | No |
| √793 | √793 | 28.1603 | No |
| √794 | √794 | 28.1780 | No |
| √795 | √795 | 28.1957 | No |
| √796 | 2√199 | 28.2135 | No |
| √797 | √797 | 28.2312 | No |
- The cube root of 794 is about 9.259911.
- Squaring undoes the root: (√794)² = 794, while 794² = 630,436 — the number whose square root is 794.
Frequently asked questions
What is the square root of 794?
The square root of 794 is √794, about 28.1780056072. The negative root, −28.178006, also squares to 794.
Is the square root of 794 rational or irrational?
Irrational. 794 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 √794 be simplified?
No. 794 = 2 × 397 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √794 rounded to two decimal places?
√794 ≈ 28.18 to two decimal places (28.2 to one, 28.178 to three). Check: 28.18² = 794.1124, close to 794.