√782 at a glance
- Exact value
- √782
- Decimal (10 places)
- 27.9642629082
- Rounded
- 28.0 · 27.96 · 27.964
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.964263
- Prime factorization
- 2 × 17 × 23
- Cube root
- 9.213025
How to simplify √782
The prime factorization of 782 is 2 × 17 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √782 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 782, 2, 17 and 23 appear an odd number of times, so √782 is irrational and 27.9642629082 is a rounded value.
Where √782 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √782 lies between 27 and 28. 782 is 53 above 729 and 2 below 784, so the root is closer to 28.
- Straight line between 729 and 784: 27.9636 (0% low)
- Tangent from 27, i.e. 27 + 53 ÷ 54: 27.9815 (0.06% high)
- Tangent from 28, i.e. 28 − 2 ÷ 56: 27.9643 (0% high)
For √782 the tangent at 28 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 782 is just 2 below 784.
Finding √782 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 | 782 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 27.9285714286 | 27.9642857143 | 4 |
| 2 | 27.9642857143 | 27.9642401022 | 27.9642629082 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √782 = 27.9642629082 to every decimal shown.
√782 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √782 the pattern is [27; 1, 26, 1, 54] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √782 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 |
|---|---|---|
| 27/1 | 27.0000000000 | 9.6 × 10⁻¹ |
| 28/1 | 28.0000000000 | 3.6 × 10⁻² |
| 755/27 | 27.9629629630 | 1.3 × 10⁻³ |
| 783/28 | 27.9642857143 | 2.3 × 10⁻⁵ |
| 43,037/1,539 | 27.9642625081 | 4.0 × 10⁻⁷ |
| 43,820/1,567 | 27.9642629228 | 1.5 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 782y² = 1. Its smallest solution in positive whole numbers is x = 783, y = 28.
√782 in geometry and everyday measurements
- 782 square feet is 72.7 m². Laid out as a square — a small house footprint or a lot — it is about 27.96 ft (28 ft) on a side.
- 782 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √782 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 7 × 27 box, because 2² + 7² + 27² = 782.
Square roots near √782 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √779 | √779 | 27.9106 | No |
| √780 | 2√195 | 27.9285 | No |
| √781 | √781 | 27.9464 | No |
| √782 | √782 | 27.9643 | No |
| √783 | 3√87 | 27.9821 | No |
| √784 | 28 | 28.0000 | Yes |
| √785 | √785 | 28.0179 | No |
- The cube root of 782 is about 9.213025.
- Squaring undoes the root: (√782)² = 782, while 782² = 611,524 — the number whose square root is 782.
Frequently asked questions
What is the square root of 782?
The square root of 782 is √782, about 27.9642629082. The negative root, −27.964263, also squares to 782.
Is the square root of 782 rational or irrational?
Irrational. 782 is not a perfect square — it falls between 729 and 784 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √782 be simplified?
No. 782 = 2 × 17 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √782 rounded to two decimal places?
√782 ≈ 27.96 to two decimal places (28.0 to one, 27.964 to three). Check: 27.96² = 781.7616, close to 782.