√858 at a glance
- Exact value
- √858
- Decimal (10 places)
- 29.2916370318
- Rounded
- 29.3 · 29.29 · 29.292
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.291637
- Prime factorization
- 2 × 3 × 11 × 13
- Cube root
- 9.502308
How to simplify √858
The prime factorization of 858 is 2 × 3 × 11 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √858 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 858, 2, 3, 11 and 13 appear an odd number of times, so √858 is irrational and 29.2916370318 is a rounded value.
Where √858 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √858 lies between 29 and 30. 858 is 17 above 841 and 42 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.2881 (0.01% low)
- Tangent from 29, i.e. 29 + 17 ÷ 58: 29.2931 (0.01% high)
- Tangent from 30, i.e. 30 − 42 ÷ 60: 29.3000 (0.03% high)
For √858 the tangent at 29 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 858 is just 17 above 841.
Finding √858 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, 29 (29² = 841):
| Step | Guess x | 858 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.5862068966 | 29.2931034483 | 2 |
| 2 | 29.2931034483 | 29.2901706886 | 29.2916370685 | 7 |
| 3 | 29.2916370685 | 29.2916369950 | 29.2916370318 | 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 √858 = 29.2916370318 to every decimal shown.
√858 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √858 the pattern is [29; 3, 2, 3, 58] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √858 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 | 2.9 × 10⁻¹ |
| 88/3 | 29.3333333333 | 4.2 × 10⁻² |
| 205/7 | 29.2857142857 | 5.9 × 10⁻³ |
| 703/24 | 29.2916666667 | 3.0 × 10⁻⁵ |
| 40,979/1,399 | 29.2916368835 | 1.5 × 10⁻⁷ |
| 123,640/4,221 | 29.2916370528 | 2.1 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 858y² = 1. Its smallest solution in positive whole numbers is x = 703, y = 24.
√858 in geometry and everyday measurements
- 858 square feet is 79.7 m². Laid out as a square — a small house footprint or a lot — it is about 29.29 ft (29 ft 3 in) on a side.
- 858 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √858 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 29 box, because 1² + 4² + 29² = 858.
Square roots near √858 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √855 | 3√95 | 29.2404 | No |
| √856 | 2√214 | 29.2575 | No |
| √857 | √857 | 29.2746 | No |
| √858 | √858 | 29.2916 | No |
| √859 | √859 | 29.3087 | No |
| √860 | 2√215 | 29.3258 | No |
| √861 | √861 | 29.3428 | No |
- The cube root of 858 is about 9.502308.
- Squaring undoes the root: (√858)² = 858, while 858² = 736,164 — the number whose square root is 858.
Frequently asked questions
What is the square root of 858?
The square root of 858 is √858, about 29.2916370318. The negative root, −29.291637, also squares to 858.
Is the square root of 858 rational or irrational?
Irrational. 858 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 √858 be simplified?
No. 858 = 2 × 3 × 11 × 13 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √858 rounded to two decimal places?
√858 ≈ 29.29 to two decimal places (29.3 to one, 29.292 to three). Check: 29.29² = 857.9041, close to 858.