√789 at a glance
- Exact value
- √789
- Decimal (10 places)
- 28.0891438104
- Rounded
- 28.1 · 28.09 · 28.089
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.089144
- Prime factorization
- 3 × 263
- Cube root
- 9.240433
How to simplify √789
The prime factorization of 789 is 3 × 263. Every prime appears only once, so there is no pair to bring outside the radical — √789 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 789, 3 and 263 appear an odd number of times, so √789 is irrational and 28.0891438104 is a rounded value.
Where √789 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √789 lies between 28 and 29. 789 is 5 above 784 and 52 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.0877 (0.01% low)
- Tangent from 28, i.e. 28 + 5 ÷ 56: 28.0893 (0% high)
- Tangent from 29, i.e. 29 − 52 ÷ 58: 28.1034 (0.05% high)
For √789 the tangent at 28 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 789 is just 5 above 784.
Finding √789 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 789: following the tangent line down to zero simplifies to averaging x with 789 ÷ x.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 789 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.1785714286 | 28.0892857143 | 3 |
| 2 | 28.0892857143 | 28.0890019072 | 28.0891438107 | 9 |
| 3 | 28.0891438107 | 28.0891438100 | 28.0891438104 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √789 = 28.0891438104 to every decimal shown.
√789 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √789 the pattern is [28; 11, 4, 1, 1, 2, 3, 1, 13, 3, 1, 2, 18, …] with the block of 24 terms after the semicolon repeating forever (only the first 12 of the 24 are shown). A pattern that never ends is one more proof that √789 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 | 8.9 × 10⁻² |
| 309/11 | 28.0909090909 | 1.8 × 10⁻³ |
| 1,264/45 | 28.0888888889 | 2.5 × 10⁻⁴ |
| 1,573/56 | 28.0892857143 | 1.4 × 10⁻⁴ |
| 2,837/101 | 28.0891089109 | 3.5 × 10⁻⁵ |
| 7,247/258 | 28.0891472868 | 3.5 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 789y² = 1. Its smallest solution in positive whole numbers is x = 16,116,667,272,575, y = 573,768,548,496 — 14 digits for x, even though 789 is small, which is what makes Pell’s equation famous.
√789 in geometry and everyday measurements
- 789 square feet is 73.3 m². Laid out as a square — a small house footprint or a lot — it is about 28.09 ft (28 ft 1 in) on a side.
- 789 is not a sum of two whole-number squares — the prime factor 3 and 263 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √789 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 28 box, because 1² + 2² + 28² = 789.
Square roots near √789 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √786 | √786 | 28.0357 | No |
| √787 | √787 | 28.0535 | No |
| √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 |
- The cube root of 789 is about 9.240433.
- Squaring undoes the root: (√789)² = 789, while 789² = 622,521 — the number whose square root is 789.
Frequently asked questions
What is the square root of 789?
The square root of 789 is √789, about 28.0891438104. The negative root, −28.089144, also squares to 789.
Is the square root of 789 rational or irrational?
Irrational. 789 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 √789 be simplified?
No. 789 = 3 × 263 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √789 rounded to two decimal places?
√789 ≈ 28.09 to two decimal places (28.1 to one, 28.089 to three). Check: 28.09² = 789.0481, close to 789.