√770 at a glance
- Exact value
- √770
- Decimal (10 places)
- 27.7488738510
- Rounded
- 27.7 · 27.75 · 27.749
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.748874
- Prime factorization
- 2 × 5 × 7 × 11
- Cube root
- 9.165656
How to simplify √770
The prime factorization of 770 is 2 × 5 × 7 × 11. Every prime appears only once, so there is no pair to bring outside the radical — √770 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 770, 2, 5, 7 and 11 appear an odd number of times, so √770 is irrational and 27.7488738510 is a rounded value.
Where √770 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √770 lies between 27 and 28. 770 is 41 above 729 and 14 below 784, so the root is closer to 28.
- Straight line between 729 and 784: 27.7455 (0.01% low)
- Tangent from 27, i.e. 27 + 41 ÷ 54: 27.7593 (0.04% high)
- Tangent from 28, i.e. 28 − 14 ÷ 56: 27.7500 (0% high)
For √770 the tangent at 28 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 770 is just 14 below 784.
Finding √770 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 | 770 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 27.5000000000 | 27.7500000000 | 2 |
| 2 | 27.7500000000 | 27.7477477477 | 27.7488738739 | 7 |
| 3 | 27.7488738739 | 27.7488738282 | 27.7488738510 | 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 √770 = 27.7488738510 to every decimal shown.
√770 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √770 the pattern is [27; 1, 2, 1, 54] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √770 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 | 7.5 × 10⁻¹ |
| 28/1 | 28.0000000000 | 2.5 × 10⁻¹ |
| 83/3 | 27.6666666667 | 8.2 × 10⁻² |
| 111/4 | 27.7500000000 | 1.1 × 10⁻³ |
| 6,077/219 | 27.7488584475 | 1.5 × 10⁻⁵ |
| 6,188/223 | 27.7488789238 | 5.1 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 770y² = 1. Its smallest solution in positive whole numbers is x = 111, y = 4.
√770 in geometry and everyday measurements
- 770 square feet is 71.5 m². Laid out as a square — a small house footprint or a lot — it is about 27.75 ft (27 ft 9 in) on a side.
- 770 is not a sum of two whole-number squares — the prime factor 7 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √770 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 12 × 25 box, because 1² + 12² + 25² = 770.
Square roots near √770 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √767 | √767 | 27.6948 | No |
| √768 | 16√3 | 27.7128 | No |
| √769 | √769 | 27.7308 | No |
| √770 | √770 | 27.7489 | No |
| √771 | √771 | 27.7669 | No |
| √772 | 2√193 | 27.7849 | No |
| √773 | √773 | 27.8029 | No |
- The cube root of 770 is about 9.165656.
- Squaring undoes the root: (√770)² = 770, while 770² = 592,900 — the number whose square root is 770.
Frequently asked questions
What is the square root of 770?
The square root of 770 is √770, about 27.7488738510. The negative root, −27.748874, also squares to 770.
Is the square root of 770 rational or irrational?
Irrational. 770 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 √770 be simplified?
No. 770 = 2 × 5 × 7 × 11 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √770 rounded to two decimal places?
√770 ≈ 27.75 to two decimal places (27.7 to one, 27.749 to three). Check: 27.75² = 770.0625, close to 770.