√766 at a glance
- Exact value
- √766
- Decimal (10 places)
- 27.6767050062
- Rounded
- 27.7 · 27.68 · 27.677
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.676705
- Prime factorization
- 2 × 383
- Cube root
- 9.149758
How to simplify √766
The prime factorization of 766 is 2 × 383. Every prime appears only once, so there is no pair to bring outside the radical — √766 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 766, 2 and 383 appear an odd number of times, so √766 is irrational and 27.6767050062 is a rounded value.
Where √766 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √766 lies between 27 and 28. 766 is 37 above 729 and 18 below 784, so the root is closer to 28.
- Straight line between 729 and 784: 27.6727 (0.01% low)
- Tangent from 27, i.e. 27 + 37 ÷ 54: 27.6852 (0.03% high)
- Tangent from 28, i.e. 28 − 18 ÷ 56: 27.6786 (0.01% high)
For √766 the tangent at 28 wins, missing by only 0.0019. Tangent estimates shine when the number sits close to a perfect square — here 766 is just 18 below 784.
Finding √766 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 | 766 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 27.3571428571 | 27.6785714286 | 2 |
| 2 | 27.6785714286 | 27.6748387097 | 27.6767050691 | 7 |
| 3 | 27.6767050691 | 27.6767049433 | 27.6767050062 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √766 = 27.6767050062 to every decimal shown.
√766 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √766 the pattern is [27; 1, 2, 10, 1, 2, 1, 3, 1, 1, 17, 1, 8, …] with the block of 44 terms after the semicolon repeating forever (only the first 12 of the 44 are shown). A pattern that never ends is one more proof that √766 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 | 6.8 × 10⁻¹ |
| 28/1 | 28.0000000000 | 3.2 × 10⁻¹ |
| 83/3 | 27.6666666667 | 1.0 × 10⁻² |
| 858/31 | 27.6774193548 | 7.1 × 10⁻⁴ |
| 941/34 | 27.6764705882 | 2.3 × 10⁻⁴ |
| 2,740/99 | 27.6767676768 | 6.3 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 766y² = 1. Its smallest solution in positive whole numbers is x = 145,933,611,945,744,638,015, y = 5,272,795,728,865,625,208 — 21 digits for x, even though 766 is small, which is what makes Pell’s equation famous.
√766 in geometry and everyday measurements
- 766 square feet is 71.2 m². Laid out as a square — a small house footprint or a lot — it is about 27.68 ft (27 ft 8 in) on a side.
- 766 is not a sum of two whole-number squares — the prime factor 383 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √766 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 27 box, because 1² + 6² + 27² = 766.
Square roots near √766 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √763 | √763 | 27.6225 | No |
| √764 | 2√191 | 27.6405 | No |
| √765 | 3√85 | 27.6586 | No |
| √766 | √766 | 27.6767 | No |
| √767 | √767 | 27.6948 | No |
| √768 | 16√3 | 27.7128 | No |
| √769 | √769 | 27.7308 | No |
- The cube root of 766 is about 9.149758.
- Squaring undoes the root: (√766)² = 766, while 766² = 586,756 — the number whose square root is 766.
Frequently asked questions
What is the square root of 766?
The square root of 766 is √766, about 27.6767050062. The negative root, −27.676705, also squares to 766.
Is the square root of 766 rational or irrational?
Irrational. 766 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 √766 be simplified?
No. 766 = 2 × 383 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √766 rounded to two decimal places?
√766 ≈ 27.68 to two decimal places (27.7 to one, 27.677 to three). Check: 27.68² = 766.1824, close to 766.