√786 at a glance
- Exact value
- √786
- Decimal (10 places)
- 28.0356915378
- Rounded
- 28.0 · 28.04 · 28.036
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.035692
- Prime factorization
- 2 × 3 × 131
- Cube root
- 9.228707
How to simplify √786
The prime factorization of 786 is 2 × 3 × 131. Every prime appears only once, so there is no pair to bring outside the radical — √786 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 786, 2, 3 and 131 appear an odd number of times, so √786 is irrational and 28.0356915378 is a rounded value.
Where √786 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √786 lies between 28 and 29. 786 is 2 above 784 and 55 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.0351 (0% low)
- Tangent from 28, i.e. 28 + 2 ÷ 56: 28.0357 (0% high)
- Tangent from 29, i.e. 29 − 55 ÷ 58: 28.0517 (0.06% high)
For √786 the tangent at 28 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 786 is just 2 above 784.
Finding √786 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 | 786 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.0714285714 | 28.0357142857 | 4 |
| 2 | 28.0357142857 | 28.0356687898 | 28.0356915378 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √786 = 28.0356915378 to every decimal shown.
√786 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √786 the pattern is [28; 28, 56] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √786 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 | 3.6 × 10⁻² |
| 785/28 | 28.0357142857 | 2.3 × 10⁻⁵ |
| 43,988/1,569 | 28.0356915233 | 1.4 × 10⁻⁸ |
| 1,232,449/43,960 | 28.0356915378 | < 10⁻¹⁰ |
| 69,061,132/2,463,329 | 28.0356915378 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 786y² = 1. Its smallest solution in positive whole numbers is x = 785, y = 28.
√786 in geometry and everyday measurements
- 786 square feet is 73 m². Laid out as a square — a small house footprint or a lot — it is about 28.04 ft (28 ft) on a side.
- 786 is not a sum of two whole-number squares — the prime factor 3 and 131 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √786 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 28 box, because 1² + 1² + 28² = 786.
Square roots near √786 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √783 | 3√87 | 27.9821 | No |
| √784 | 28 | 28.0000 | Yes |
| √785 | √785 | 28.0179 | No |
| √786 | √786 | 28.0357 | No |
| √787 | √787 | 28.0535 | No |
| √788 | 2√197 | 28.0713 | No |
| √789 | √789 | 28.0891 | No |
- The cube root of 786 is about 9.228707.
- Squaring undoes the root: (√786)² = 786, while 786² = 617,796 — the number whose square root is 786.
Frequently asked questions
What is the square root of 786?
The square root of 786 is √786, about 28.0356915378. The negative root, −28.035692, also squares to 786.
Is the square root of 786 rational or irrational?
Irrational. 786 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 √786 be simplified?
No. 786 = 2 × 3 × 131 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √786 rounded to two decimal places?
√786 ≈ 28.04 to two decimal places (28.0 to one, 28.036 to three). Check: 28.04² = 786.2416, close to 786.