√798 at a glance
- Exact value
- √798
- Decimal (10 places)
- 28.2488937837
- Rounded
- 28.2 · 28.25 · 28.249
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.248894
- Prime factorization
- 2 × 3 × 7 × 19
- Cube root
- 9.275435
How to simplify √798
The prime factorization of 798 is 2 × 3 × 7 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √798 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 798, 2, 3, 7 and 19 appear an odd number of times, so √798 is irrational and 28.2488937837 is a rounded value.
Where √798 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √798 lies between 28 and 29. 798 is 14 above 784 and 43 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.2456 (0.01% low)
- Tangent from 28, i.e. 28 + 14 ÷ 56: 28.2500 (0% high)
- Tangent from 29, i.e. 29 − 43 ÷ 58: 28.2586 (0.03% high)
For √798 the tangent at 28 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 798 is just 14 above 784.
Finding √798 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 | 798 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.5000000000 | 28.2500000000 | 2 |
| 2 | 28.2500000000 | 28.2477876106 | 28.2488938053 | 7 |
| 3 | 28.2488938053 | 28.2488937620 | 28.2488937837 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √798 = 28.2488937837 to every decimal shown.
√798 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √798 the pattern is [28; 4, 56] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √798 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 | 2.5 × 10⁻¹ |
| 113/4 | 28.2500000000 | 1.1 × 10⁻³ |
| 6,356/225 | 28.2488888889 | 4.9 × 10⁻⁶ |
| 25,537/904 | 28.2488938053 | 2.2 × 10⁻⁸ |
| 1,436,428/50,849 | 28.2488937836 | 9.6 × 10⁻¹¹ |
| 5,771,249/204,300 | 28.2488937837 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 798y² = 1. Its smallest solution in positive whole numbers is x = 113, y = 4.
√798 in geometry and everyday measurements
- 798 square feet is 74.1 m². Laid out as a square — a small house footprint or a lot — it is about 28.25 ft (28 ft 3 in) on a side.
- 798 is not a sum of two whole-number squares — the prime factor 3, 7 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √798 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 11 × 26 box, because 1² + 11² + 26² = 798.
Square roots near √798 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √795 | √795 | 28.1957 | No |
| √796 | 2√199 | 28.2135 | No |
| √797 | √797 | 28.2312 | No |
| √798 | √798 | 28.2489 | No |
| √799 | √799 | 28.2666 | No |
| √800 | 20√2 | 28.2843 | No |
| √801 | 3√89 | 28.3019 | No |
- The cube root of 798 is about 9.275435.
- Squaring undoes the root: (√798)² = 798, while 798² = 636,804 — the number whose square root is 798.
Frequently asked questions
What is the square root of 798?
The square root of 798 is √798, about 28.2488937837. The negative root, −28.248894, also squares to 798.
Is the square root of 798 rational or irrational?
Irrational. 798 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 √798 be simplified?
No. 798 = 2 × 3 × 7 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √798 rounded to two decimal places?
√798 ≈ 28.25 to two decimal places (28.2 to one, 28.249 to three). Check: 28.25² = 798.0625, close to 798.