√785 at a glance
- Exact value
- √785
- Decimal (10 places)
- 28.0178514522
- Rounded
- 28.0 · 28.02 · 28.018
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.017851
- Prime factorization
- 5 × 157
- Cube root
- 9.224791
How to simplify √785
The prime factorization of 785 is 5 × 157. Every prime appears only once, so there is no pair to bring outside the radical — √785 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 785, 5 and 157 appear an odd number of times, so √785 is irrational and 28.0178514522 is a rounded value.
Where √785 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √785 lies between 28 and 29. 785 is 1 above 784 and 56 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.0175 (0% low)
- Tangent from 28, i.e. 28 + 1 ÷ 56: 28.0179 (0% high)
- Tangent from 29, i.e. 29 − 56 ÷ 58: 28.0345 (0.06% high)
For √785 the tangent at 28 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 785 is just 1 above 784.
Finding √785 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 785: following the tangent line down to zero simplifies to averaging x with 785 ÷ x.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 785 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.0357142857 | 28.0178571429 | 5 |
| 2 | 28.0178571429 | 28.0178457616 | 28.0178514522 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √785 = 28.0178514522 to every decimal shown.
√785 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √785 the pattern is [28; 56] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 785 is one more than a perfect square (28² + 1). A pattern that never ends is one more proof that √785 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 | 1.8 × 10⁻² |
| 1,569/56 | 28.0178571429 | 5.7 × 10⁻⁶ |
| 87,892/3,137 | 28.0178514504 | 1.8 × 10⁻⁹ |
| 4,923,521/175,728 | 28.0178514522 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 785y² = 1. Its smallest solution in positive whole numbers is x = 1,569, y = 56. Because the period is odd, the equation with −1 on the right also has a solution: 28² − 785 × 1² = −1.
√785 in geometry and everyday measurements
- 785 square feet is 72.9 m². Laid out as a square — a small house footprint or a lot — it is about 28.02 ft (28 ft) on a side.
- 785 = 1² + 28² = 16² + 23², so by the Pythagorean theorem √785 is the diagonal of rectangles measuring 1 × 28 and 16 × 23 — and the distance between the points (0, 0) and (1, 28) on a grid.
Square roots near √785 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √782 | √782 | 27.9643 | No |
| √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 |
- The cube root of 785 is about 9.224791.
- Squaring undoes the root: (√785)² = 785, while 785² = 616,225 — the number whose square root is 785.
Frequently asked questions
What is the square root of 785?
The square root of 785 is √785, about 28.0178514522. The negative root, −28.017851, also squares to 785.
Is the square root of 785 rational or irrational?
Irrational. 785 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 √785 be simplified?
No. 785 = 5 × 157 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √785 rounded to two decimal places?
√785 ≈ 28.02 to two decimal places (28.0 to one, 28.018 to three). Check: 28.02² = 785.1204, close to 785.