√781 at a glance
- Exact value
- √781
- Decimal (10 places)
- 27.9463772250
- Rounded
- 27.9 · 27.95 · 27.946
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.946377
- Prime factorization
- 11 × 71
- Cube root
- 9.209096
How to simplify √781
The prime factorization of 781 is 11 × 71. Every prime appears only once, so there is no pair to bring outside the radical — √781 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 781, 11 and 71 appear an odd number of times, so √781 is irrational and 27.9463772250 is a rounded value.
Where √781 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √781 lies between 27 and 28. 781 is 52 above 729 and 3 below 784, so the root is closer to 28.
- Straight line between 729 and 784: 27.9455 (0% low)
- Tangent from 27, i.e. 27 + 52 ÷ 54: 27.9630 (0.06% high)
- Tangent from 28, i.e. 28 − 3 ÷ 56: 27.9464 (0% high)
For √781 the tangent at 28 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 781 is just 3 below 784.
Finding √781 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 781: following the tangent line down to zero simplifies to averaging x with 781 ÷ x.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 781 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 27.8928571429 | 27.9464285714 | 4 |
| 2 | 27.9464285714 | 27.9463258786 | 27.9463772250 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √781 = 27.9463772250 to every decimal shown.
√781 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √781 the pattern is [27; 1, 17, 1, 1, 1, 5, 1, 1, 4, 1, 1, 5, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √781 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 | 9.5 × 10⁻¹ |
| 28/1 | 28.0000000000 | 5.4 × 10⁻² |
| 503/18 | 27.9444444444 | 1.9 × 10⁻³ |
| 531/19 | 27.9473684211 | 9.9 × 10⁻⁴ |
| 1,034/37 | 27.9459459459 | 4.3 × 10⁻⁴ |
| 1,565/56 | 27.9464285714 | 5.1 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 781y² = 1. Its smallest solution in positive whole numbers is x = 67,606,199, y = 2,419,140.
√781 in geometry and everyday measurements
- 781 square feet is 72.6 m². Laid out as a square — a small house footprint or a lot — it is about 27.95 ft (27 ft 11 in) on a side.
- 781 is not a sum of two whole-number squares — the prime factor 11 and 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √781 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 14 × 24 box, because 3² + 14² + 24² = 781.
Square roots near √781 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √778 | √778 | 27.8927 | No |
| √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 |
- The cube root of 781 is about 9.209096.
- Squaring undoes the root: (√781)² = 781, while 781² = 609,961 — the number whose square root is 781.
Frequently asked questions
What is the square root of 781?
The square root of 781 is √781, about 27.9463772250. The negative root, −27.946377, also squares to 781.
Is the square root of 781 rational or irrational?
Irrational. 781 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 √781 be simplified?
No. 781 = 11 × 71 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √781 rounded to two decimal places?
√781 ≈ 27.95 to two decimal places (27.9 to one, 27.946 to three). Check: 27.95² = 781.2025, close to 781.