√886 at a glance
- Exact value
- √886
- Decimal (10 places)
- 29.7657521323
- Rounded
- 29.8 · 29.77 · 29.766
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.765752
- Prime factorization
- 2 × 443
- Cube root
- 9.604570
How to simplify √886
The prime factorization of 886 is 2 × 443. Every prime appears only once, so there is no pair to bring outside the radical — √886 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 886, 2 and 443 appear an odd number of times, so √886 is irrational and 29.7657521323 is a rounded value.
Where √886 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √886 lies between 29 and 30. 886 is 45 above 841 and 14 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.7627 (0.01% low)
- Tangent from 29, i.e. 29 + 45 ÷ 58: 29.7759 (0.03% high)
- Tangent from 30, i.e. 30 − 14 ÷ 60: 29.7667 (0% high)
For √886 the tangent at 30 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 886 is just 14 below 900.
Finding √886 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, 30 (30² = 900):
| Step | Guess x | 886 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.5333333333 | 29.7666666667 | 3 |
| 2 | 29.7666666667 | 29.7648376260 | 29.7657521463 | 7 |
| 3 | 29.7657521463 | 29.7657521182 | 29.7657521323 | 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 √886 = 29.7657521323 to every decimal shown.
√886 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √886 the pattern is [29; 1, 3, 3, 1, 2, 1, 1, 5, 2, 1, 1, 1, …] with the block of 54 terms after the semicolon repeating forever (only the first 12 of the 54 are shown). A pattern that never ends is one more proof that √886 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 | 7.7 × 10⁻¹ |
| 30/1 | 30.0000000000 | 2.3 × 10⁻¹ |
| 119/4 | 29.7500000000 | 1.6 × 10⁻² |
| 387/13 | 29.7692307692 | 3.5 × 10⁻³ |
| 506/17 | 29.7647058824 | 1.0 × 10⁻³ |
| 1,399/47 | 29.7659574468 | 2.1 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 886y² = 1. Its smallest solution in positive whole numbers is x = 7,743,524,593,057,655,851,637,765, y = 260,148,796,464,024,194,850,378 — 25 digits for x, even though 886 is small, which is what makes Pell’s equation famous.
√886 in geometry and everyday measurements
- 886 square feet is 82.3 m². Laid out as a square — a small house footprint or a lot — it is about 29.77 ft (29 ft 9 in) on a side.
- 886 is not a sum of two whole-number squares — the prime factor 443 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √886 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 21 × 21 box, because 2² + 21² + 21² = 886.
Square roots near √886 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √883 | √883 | 29.7153 | No |
| √884 | 2√221 | 29.7321 | No |
| √885 | √885 | 29.7489 | No |
| √886 | √886 | 29.7658 | No |
| √887 | √887 | 29.7825 | No |
| √888 | 2√222 | 29.7993 | No |
| √889 | √889 | 29.8161 | No |
- The cube root of 886 is about 9.604570.
- Squaring undoes the root: (√886)² = 886, while 886² = 784,996 — the number whose square root is 886.
Frequently asked questions
What is the square root of 886?
The square root of 886 is √886, about 29.7657521323. The negative root, −29.765752, also squares to 886.
Is the square root of 886 rational or irrational?
Irrational. 886 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 √886 be simplified?
No. 886 = 2 × 443 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √886 rounded to two decimal places?
√886 ≈ 29.77 to two decimal places (29.8 to one, 29.766 to three). Check: 29.77² = 886.2529, close to 886.