√797 at a glance
- Exact value
- √797
- Decimal (10 places)
- 28.2311884270
- Rounded
- 28.2 · 28.23 · 28.231
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.231188
- Prime factorization
- 797
- Cube root
- 9.271559
How to simplify √797
797 is a prime number, so its only factors are 1 and 797. There is no perfect-square factor to pull out, which means √797 is already in its simplest radical form.
The square root of any prime is irrational. If √797 were a fraction a/b in lowest terms, then a² = 797b², so 797 would divide a — and then 797 would divide b too, contradicting “lowest terms.” That is why the decimal 28.2311884270 is only a rounded value.
Where √797 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √797 lies between 28 and 29. 797 is 13 above 784 and 44 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.2281 (0.01% low)
- Tangent from 28, i.e. 28 + 13 ÷ 56: 28.2321 (0% high)
- Tangent from 29, i.e. 29 − 44 ÷ 58: 28.2414 (0.04% high)
For √797 the tangent at 28 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 797 is just 13 above 784.
Finding √797 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 797: following the tangent line down to zero simplifies to averaging x with 797 ÷ x.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 797 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.4642857143 | 28.2321428571 | 3 |
| 2 | 28.2321428571 | 28.2302340291 | 28.2311884431 | 7 |
| 3 | 28.2311884431 | 28.2311884109 | 28.2311884270 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √797 = 28.2311884270 to every decimal shown.
√797 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √797 the pattern is [28; 4, 3, 13, 1, 4, 4, 1, 13, 3, 4, 56] with the block of 11 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √797 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 | 2.3 × 10⁻¹ |
| 113/4 | 28.2500000000 | 1.9 × 10⁻² |
| 367/13 | 28.2307692308 | 4.2 × 10⁻⁴ |
| 4,884/173 | 28.2312138728 | 2.5 × 10⁻⁵ |
| 5,251/186 | 28.2311827957 | 5.6 × 10⁻⁶ |
| 25,888/917 | 28.2311886587 | 2.3 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 797y² = 1. Its smallest solution in positive whole numbers is x = 1,221,759,532,448,649, y = 43,276,943,002,540 — 16 digits for x, even though 797 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 24,715,982² − 797 × 875,485² = −1.
√797 in geometry and everyday measurements
- 797 square feet is 74 m². Laid out as a square — a small house footprint or a lot — it is about 28.23 ft (28 ft 3 in) on a side.
- 797 = 11² + 26², so by the Pythagorean theorem √797 is the diagonal of a 11 × 26 rectangle — and the distance between the points (0, 0) and (11, 26) on a grid.
Square roots near √797 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √794 | √794 | 28.1780 | No |
| √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 |
- The cube root of 797 is about 9.271559.
- Squaring undoes the root: (√797)² = 797, while 797² = 635,209 — the number whose square root is 797.
Frequently asked questions
What is the square root of 797?
The square root of 797 is √797, about 28.2311884270. The negative root, −28.231188, also squares to 797.
Is the square root of 797 rational or irrational?
Irrational. 797 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 √797 be simplified?
No. 797 is prime, so there is no perfect square to take out of the radical.
What is √797 rounded to two decimal places?
√797 ≈ 28.23 to two decimal places (28.2 to one, 28.231 to three). Check: 28.23² = 796.9329, close to 797.