√791 at a glance
- Exact value
- √791
- Decimal (10 places)
- 28.1247222209
- Rounded
- 28.1 · 28.12 · 28.125
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.124722
- Prime factorization
- 7 × 113
- Cube root
- 9.248234
How to simplify √791
The prime factorization of 791 is 7 × 113. Every prime appears only once, so there is no pair to bring outside the radical — √791 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 791, 7 and 113 appear an odd number of times, so √791 is irrational and 28.1247222209 is a rounded value.
Where √791 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √791 lies between 28 and 29. 791 is 7 above 784 and 50 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.1228 (0.01% low)
- Tangent from 28, i.e. 28 + 7 ÷ 56: 28.1250 (0% high)
- Tangent from 29, i.e. 29 − 50 ÷ 58: 28.1379 (0.05% high)
For √791 the tangent at 28 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 791 is just 7 above 784.
Finding √791 with the Babylonian method
If a guess is too big, 791 ÷ 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√791) in one step.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 791 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.2500000000 | 28.1250000000 | 3 |
| 2 | 28.1250000000 | 28.1244444444 | 28.1247222222 | 8 |
| 3 | 28.1247222222 | 28.1247222195 | 28.1247222209 | 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 √791 = 28.1247222209 to every decimal shown.
√791 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √791 the pattern is [28; 8, 56] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √791 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 | 1.2 × 10⁻¹ |
| 225/8 | 28.1250000000 | 2.8 × 10⁻⁴ |
| 12,628/449 | 28.1247216036 | 6.2 × 10⁻⁷ |
| 101,249/3,600 | 28.1247222222 | 1.4 × 10⁻⁹ |
| 5,682,572/202,049 | 28.1247222208 | < 10⁻¹⁰ |
| 45,561,825/1,619,992 | 28.1247222209 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 791y² = 1. Its smallest solution in positive whole numbers is x = 225, y = 8.
√791 in geometry and everyday measurements
- 791 square feet is 73.5 m². Laid out as a square — a small house footprint or a lot — it is about 28.12 ft (28 ft 1 in) on a side.
- 791 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √791 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 √791 as its space diagonal.
Square roots near √791 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √794 | √794 | 28.1780 | No |
- The cube root of 791 is about 9.248234.
- Squaring undoes the root: (√791)² = 791, while 791² = 625,681 — the number whose square root is 791.
Frequently asked questions
What is the square root of 791?
The square root of 791 is √791, about 28.1247222209. The negative root, −28.124722, also squares to 791.
Is the square root of 791 rational or irrational?
Irrational. 791 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 √791 be simplified?
No. 791 = 7 × 113 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √791 rounded to two decimal places?
√791 ≈ 28.12 to two decimal places (28.1 to one, 28.125 to three). Check: 28.12² = 790.7344, close to 791.