√790 at a glance
- Exact value
- √790
- Decimal (10 places)
- 28.1069386451
- Rounded
- 28.1 · 28.11 · 28.107
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.106939
- Prime factorization
- 2 × 5 × 79
- Cube root
- 9.244335
How to simplify √790
The prime factorization of 790 is 2 × 5 × 79. Every prime appears only once, so there is no pair to bring outside the radical — √790 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 790, 2, 5 and 79 appear an odd number of times, so √790 is irrational and 28.1069386451 is a rounded value.
Where √790 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √790 lies between 28 and 29. 790 is 6 above 784 and 51 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.1053 (0.01% low)
- Tangent from 28, i.e. 28 + 6 ÷ 56: 28.1071 (0% high)
- Tangent from 29, i.e. 29 − 51 ÷ 58: 28.1207 (0.05% high)
For √790 the tangent at 28 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 790 is just 6 above 784.
Finding √790 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 | 790 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.2142857143 | 28.1071428571 | 3 |
| 2 | 28.1071428571 | 28.1067344346 | 28.1069386459 | 9 |
| 3 | 28.1069386459 | 28.1069386444 | 28.1069386451 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √790 = 28.1069386451 to every decimal shown.
√790 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √790 the pattern is [28; 9, 2, 1, 5, 1, 1, 3, 4, 1, 4, 1, 4, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √790 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 | 1.1 × 10⁻¹ |
| 253/9 | 28.1111111111 | 4.2 × 10⁻³ |
| 534/19 | 28.1052631579 | 1.7 × 10⁻³ |
| 787/28 | 28.1071428571 | 2.0 × 10⁻⁴ |
| 4,469/159 | 28.1069182390 | 2.0 × 10⁻⁵ |
| 5,256/187 | 28.1069518717 | 1.3 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 790y² = 1. Its smallest solution in positive whole numbers is x = 6,616,066,879, y = 235,389,096.
√790 in geometry and everyday measurements
- 790 square feet is 73.4 m². Laid out as a square — a small house footprint or a lot — it is about 28.11 ft (28 ft 1 in) on a side.
- 790 is not a sum of two whole-number squares — the prime factor 79 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √790 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 6 × 27 box, because 5² + 6² + 27² = 790.
Square roots near √790 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √787 | √787 | 28.0535 | No |
| √788 | 2√197 | 28.0713 | No |
| √789 | √789 | 28.0891 | No |
| √790 | √790 | 28.1069 | No |
| √791 | √791 | 28.1247 | No |
| √792 | 6√22 | 28.1425 | No |
| √793 | √793 | 28.1603 | No |
- The cube root of 790 is about 9.244335.
- Squaring undoes the root: (√790)² = 790, while 790² = 624,100 — the number whose square root is 790.
Frequently asked questions
What is the square root of 790?
The square root of 790 is √790, about 28.1069386451. The negative root, −28.106939, also squares to 790.
Is the square root of 790 rational or irrational?
Irrational. 790 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 √790 be simplified?
No. 790 = 2 × 5 × 79 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √790 rounded to two decimal places?
√790 ≈ 28.11 to two decimal places (28.1 to one, 28.107 to three). Check: 28.11² = 790.1721, close to 790.