√767 at a glance
- Exact value
- √767
- Decimal (10 places)
- 27.6947648483
- Rounded
- 27.7 · 27.69 · 27.695
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.694765
- Prime factorization
- 13 × 59
- Cube root
- 9.153738
How to simplify √767
The prime factorization of 767 is 13 × 59. Every prime appears only once, so there is no pair to bring outside the radical — √767 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 767, 13 and 59 appear an odd number of times, so √767 is irrational and 27.6947648483 is a rounded value.
Where √767 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √767 lies between 27 and 28. 767 is 38 above 729 and 17 below 784, so the root is closer to 28.
- Straight line between 729 and 784: 27.6909 (0.01% low)
- Tangent from 27, i.e. 27 + 38 ÷ 54: 27.7037 (0.03% high)
- Tangent from 28, i.e. 28 − 17 ÷ 56: 27.6964 (0.01% high)
For √767 the tangent at 28 wins, missing by only 0.0017. Tangent estimates shine when the number sits close to a perfect square — here 767 is just 17 below 784.
Finding √767 with the Babylonian method
If a guess is too big, 767 ÷ 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√767) in one step.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 767 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 27.3928571429 | 27.6964285714 | 2 |
| 2 | 27.6964285714 | 27.6931012250 | 27.6947648982 | 7 |
| 3 | 27.6947648982 | 27.6947647983 | 27.6947648483 | 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 √767 = 27.6947648483 to every decimal shown.
√767 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √767 the pattern is [27; 1, 2, 3, 1, 1, 1, 1, 1, 3, 2, 1, 54] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √767 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 | 6.9 × 10⁻¹ |
| 28/1 | 28.0000000000 | 3.1 × 10⁻¹ |
| 83/3 | 27.6666666667 | 2.8 × 10⁻² |
| 277/10 | 27.7000000000 | 5.2 × 10⁻³ |
| 360/13 | 27.6923076923 | 2.5 × 10⁻³ |
| 637/23 | 27.6956521739 | 8.9 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 767y² = 1. Its smallest solution in positive whole numbers is x = 31,212, y = 1,127.
√767 in geometry and everyday measurements
- 767 square feet is 71.3 m². Laid out as a square — a small house footprint or a lot — it is about 27.69 ft (27 ft 8 in) on a side.
- 767 is not a sum of two whole-number squares — the prime factor 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √767 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √767 as its space diagonal.
Square roots near √767 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √770 | √770 | 27.7489 | No |
- The cube root of 767 is about 9.153738.
- Squaring undoes the root: (√767)² = 767, while 767² = 588,289 — the number whose square root is 767.
Frequently asked questions
What is the square root of 767?
The square root of 767 is √767, about 27.6947648483. The negative root, −27.694765, also squares to 767.
Is the square root of 767 rational or irrational?
Irrational. 767 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 √767 be simplified?
No. 767 = 13 × 59 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √767 rounded to two decimal places?
√767 ≈ 27.69 to two decimal places (27.7 to one, 27.695 to three). Check: 27.69² = 766.7361, close to 767.