√849 at a glance
- Exact value
- √849
- Decimal (10 places)
- 29.1376045687
- Rounded
- 29.1 · 29.14 · 29.138
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.137605
- Prime factorization
- 3 × 283
- Cube root
- 9.468966
How to simplify √849
The prime factorization of 849 is 3 × 283. Every prime appears only once, so there is no pair to bring outside the radical — √849 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 849, 3 and 283 appear an odd number of times, so √849 is irrational and 29.1376045687 is a rounded value.
Where √849 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √849 lies between 29 and 30. 849 is 8 above 841 and 51 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.1356 (0.01% low)
- Tangent from 29, i.e. 29 + 8 ÷ 58: 29.1379 (0% high)
- Tangent from 30, i.e. 30 − 51 ÷ 60: 29.1500 (0.04% high)
For √849 the tangent at 29 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 849 is just 8 above 841.
Finding √849 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 849: following the tangent line down to zero simplifies to averaging x with 849 ÷ x.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 849 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.2758620690 | 29.1379310345 | 3 |
| 2 | 29.1379310345 | 29.1372781065 | 29.1376045705 | 8 |
| 3 | 29.1376045705 | 29.1376045668 | 29.1376045687 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √849 = 29.1376045687 to every decimal shown.
√849 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √849 the pattern is [29; 7, 3, 1, 2, 1, 7, 1, 1, 2, 4, 11, 2, …] with the block of 30 terms after the semicolon repeating forever (only the first 12 of the 30 are shown). A pattern that never ends is one more proof that √849 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 |
|---|---|---|
| 29/1 | 29.0000000000 | 1.4 × 10⁻¹ |
| 204/7 | 29.1428571429 | 5.3 × 10⁻³ |
| 641/22 | 29.1363636364 | 1.2 × 10⁻³ |
| 845/29 | 29.1379310345 | 3.3 × 10⁻⁴ |
| 2,331/80 | 29.1375000000 | 1.0 × 10⁻⁴ |
| 3,176/109 | 29.1376146789 | 1.0 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 849y² = 1. Its smallest solution in positive whole numbers is x = 1,501,654,712,948,695, y = 51,536,656,330,476 — 16 digits for x, even though 849 is small, which is what makes Pell’s equation famous.
√849 in geometry and everyday measurements
- 849 square feet is 78.9 m². Laid out as a square — a small house footprint or a lot — it is about 29.14 ft (29 ft 2 in) on a side.
- 849 is not a sum of two whole-number squares — the prime factor 3 and 283 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √849 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 8 × 28 box, because 1² + 8² + 28² = 849.
Square roots near √849 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √846 | 3√94 | 29.0861 | No |
| √847 | 11√7 | 29.1033 | No |
| √848 | 4√53 | 29.1204 | No |
| √849 | √849 | 29.1376 | No |
| √850 | 5√34 | 29.1548 | No |
| √851 | √851 | 29.1719 | No |
| √852 | 2√213 | 29.1890 | No |
- The cube root of 849 is about 9.468966.
- Squaring undoes the root: (√849)² = 849, while 849² = 720,801 — the number whose square root is 849.
Frequently asked questions
What is the square root of 849?
The square root of 849 is √849, about 29.1376045687. The negative root, −29.137605, also squares to 849.
Is the square root of 849 rational or irrational?
Irrational. 849 is not a perfect square — it falls between 841 and 900 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √849 be simplified?
No. 849 = 3 × 283 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √849 rounded to two decimal places?
√849 ≈ 29.14 to two decimal places (29.1 to one, 29.138 to three). Check: 29.14² = 849.1396, close to 849.