√799 at a glance
- Exact value
- √799
- Decimal (10 places)
- 28.2665880502
- Rounded
- 28.3 · 28.27 · 28.267
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.266588
- Prime factorization
- 17 × 47
- Cube root
- 9.279308
How to simplify √799
The prime factorization of 799 is 17 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √799 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 799, 17 and 47 appear an odd number of times, so √799 is irrational and 28.2665880502 is a rounded value.
Where √799 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √799 lies between 28 and 29. 799 is 15 above 784 and 42 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.2632 (0.01% low)
- Tangent from 28, i.e. 28 + 15 ÷ 56: 28.2679 (0% high)
- Tangent from 29, i.e. 29 − 42 ÷ 58: 28.2759 (0.03% high)
For √799 the tangent at 28 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 799 is just 15 above 784.
Finding √799 with the Babylonian method
If a guess is too big, 799 ÷ 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√799) in one step.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 799 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.5357142857 | 28.2678571429 | 2 |
| 2 | 28.2678571429 | 28.2653190145 | 28.2665880787 | 7 |
| 3 | 28.2665880787 | 28.2665880217 | 28.2665880502 | 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 √799 = 28.2665880502 to every decimal shown.
√799 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √799 the pattern is [28; 3, 1, 3, 56] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √799 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 | 2.7 × 10⁻¹ |
| 85/3 | 28.3333333333 | 6.7 × 10⁻² |
| 113/4 | 28.2500000000 | 1.7 × 10⁻² |
| 424/15 | 28.2666666667 | 7.9 × 10⁻⁵ |
| 23,857/844 | 28.2665876777 | 3.7 × 10⁻⁷ |
| 71,995/2,547 | 28.2665881429 | 9.3 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 799y² = 1. Its smallest solution in positive whole numbers is x = 424, y = 15.
√799 in geometry and everyday measurements
- 799 square feet is 74.2 m². Laid out as a square — a small house footprint or a lot — it is about 28.27 ft (28 ft 3 in) on a side.
- 799 is not a sum of two whole-number squares — the prime factor 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √799 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 √799 as its space diagonal.
Square roots near √799 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √802 | √802 | 28.3196 | No |
- The cube root of 799 is about 9.279308.
- Squaring undoes the root: (√799)² = 799, while 799² = 638,401 — the number whose square root is 799.
Frequently asked questions
What is the square root of 799?
The square root of 799 is √799, about 28.2665880502. The negative root, −28.266588, also squares to 799.
Is the square root of 799 rational or irrational?
Irrational. 799 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 √799 be simplified?
No. 799 = 17 × 47 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √799 rounded to two decimal places?
√799 ≈ 28.27 to two decimal places (28.3 to one, 28.267 to three). Check: 28.27² = 799.1929, close to 799.