√793 at a glance
- Exact value
- √793
- Decimal (10 places)
- 28.1602556807
- Rounded
- 28.2 · 28.16 · 28.160
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.160256
- Prime factorization
- 13 × 61
- Cube root
- 9.256022
How to simplify √793
The prime factorization of 793 is 13 × 61. Every prime appears only once, so there is no pair to bring outside the radical — √793 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 793, 13 and 61 appear an odd number of times, so √793 is irrational and 28.1602556807 is a rounded value.
Where √793 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √793 lies between 28 and 29. 793 is 9 above 784 and 48 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.1579 (0.01% low)
- Tangent from 28, i.e. 28 + 9 ÷ 56: 28.1607 (0% high)
- Tangent from 29, i.e. 29 − 48 ÷ 58: 28.1724 (0.04% high)
For √793 the tangent at 28 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 793 is just 9 above 784.
Finding √793 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 793: following the tangent line down to zero simplifies to averaging x with 793 ÷ x.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 793 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.3214285714 | 28.1607142857 | 3 |
| 2 | 28.1607142857 | 28.1597970831 | 28.1602556844 | 8 |
| 3 | 28.1602556844 | 28.1602556769 | 28.1602556807 | 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 √793 = 28.1602556807 to every decimal shown.
√793 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √793 the pattern is [28; 6, 4, 6, 56] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √793 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 | 1.6 × 10⁻¹ |
| 169/6 | 28.1666666667 | 6.4 × 10⁻³ |
| 704/25 | 28.1600000000 | 2.6 × 10⁻⁴ |
| 4,393/156 | 28.1602564103 | 7.3 × 10⁻⁷ |
| 246,712/8,761 | 28.1602556786 | 2.1 × 10⁻⁹ |
| 1,484,665/52,722 | 28.1602556807 | 8.3 × 10⁻¹¹ |
The same fractions solve Pell’s equation, x² − 793y² = 1. Its smallest solution in positive whole numbers is x = 4,393, y = 156.
√793 in geometry and everyday measurements
- 793 square feet is 73.7 m². Laid out as a square — a small house footprint or a lot — it is about 28.16 ft (28 ft 2 in) on a side.
- 793 = 3² + 28² = 8² + 27², so by the Pythagorean theorem √793 is the diagonal of rectangles measuring 3 × 28 and 8 × 27 — and the distance between the points (0, 0) and (3, 28) on a grid.
Square roots near √793 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √795 | √795 | 28.1957 | No |
| √796 | 2√199 | 28.2135 | No |
- The cube root of 793 is about 9.256022.
- Squaring undoes the root: (√793)² = 793, while 793² = 628,849 — the number whose square root is 793.
Frequently asked questions
What is the square root of 793?
The square root of 793 is √793, about 28.1602556807. The negative root, −28.160256, also squares to 793.
Is the square root of 793 rational or irrational?
Irrational. 793 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 √793 be simplified?
No. 793 = 13 × 61 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √793 rounded to two decimal places?
√793 ≈ 28.16 to two decimal places (28.2 to one, 28.160 to three). Check: 28.16² = 792.9856, close to 793.