√771 at a glance
- Exact value
- √771
- Decimal (10 places)
- 27.7668867538
- Rounded
- 27.8 · 27.77 · 27.767
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.766887
- Prime factorization
- 3 × 257
- Cube root
- 9.169623
How to simplify √771
The prime factorization of 771 is 3 × 257. Every prime appears only once, so there is no pair to bring outside the radical — √771 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 771, 3 and 257 appear an odd number of times, so √771 is irrational and 27.7668867538 is a rounded value.
Where √771 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √771 lies between 27 and 28. 771 is 42 above 729 and 13 below 784, so the root is closer to 28.
- Straight line between 729 and 784: 27.7636 (0.01% low)
- Tangent from 27, i.e. 27 + 42 ÷ 54: 27.7778 (0.04% high)
- Tangent from 28, i.e. 28 − 13 ÷ 56: 27.7679 (0% high)
For √771 the tangent at 28 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 771 is just 13 below 784.
Finding √771 with the Babylonian method
If a guess is too big, 771 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√771) in one step.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 771 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 27.5357142857 | 27.7678571429 | 3 |
| 2 | 27.7678571429 | 27.7659163987 | 27.7668867708 | 7 |
| 3 | 27.7668867708 | 27.7668867369 | 27.7668867538 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √771 = 27.7668867538 to every decimal shown.
√771 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √771 the pattern is [27; 1, 3, 3, 2, 4, 1, 1, 1, 1, 2, 27, 2, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √771 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 | 7.7 × 10⁻¹ |
| 28/1 | 28.0000000000 | 2.3 × 10⁻¹ |
| 111/4 | 27.7500000000 | 1.7 × 10⁻² |
| 361/13 | 27.7692307692 | 2.3 × 10⁻³ |
| 833/30 | 27.7666666667 | 2.2 × 10⁻⁴ |
| 3,693/133 | 27.7669172932 | 3.1 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 771y² = 1. Its smallest solution in positive whole numbers is x = 2,989,136,930, y = 107,651,137.
√771 in geometry and everyday measurements
- 771 square feet is 71.6 m². Laid out as a square — a small house footprint or a lot — it is about 27.77 ft (27 ft 9 in) on a side.
- 771 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √771 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 11 × 25 box, because 5² + 11² + 25² = 771.
Square roots near √771 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √774 | 3√86 | 27.8209 | No |
- The cube root of 771 is about 9.169623.
- Squaring undoes the root: (√771)² = 771, while 771² = 594,441 — the number whose square root is 771.
Frequently asked questions
What is the square root of 771?
The square root of 771 is √771, about 27.7668867538. The negative root, −27.766887, also squares to 771.
Is the square root of 771 rational or irrational?
Irrational. 771 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 √771 be simplified?
No. 771 = 3 × 257 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √771 rounded to two decimal places?
√771 ≈ 27.77 to two decimal places (27.8 to one, 27.767 to three). Check: 27.77² = 771.1729, close to 771.