√862 at a glance
- Exact value
- √862
- Decimal (10 places)
- 29.3598365118
- Rounded
- 29.4 · 29.36 · 29.360
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.359837
- Prime factorization
- 2 × 431
- Cube root
- 9.517052
How to simplify √862
The prime factorization of 862 is 2 × 431. Every prime appears only once, so there is no pair to bring outside the radical — √862 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 862, 2 and 431 appear an odd number of times, so √862 is irrational and 29.3598365118 is a rounded value.
Where √862 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √862 lies between 29 and 30. 862 is 21 above 841 and 38 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.3559 (0.01% low)
- Tangent from 29, i.e. 29 + 21 ÷ 58: 29.3621 (0.01% high)
- Tangent from 30, i.e. 30 − 38 ÷ 60: 29.3667 (0.02% high)
For √862 the tangent at 29 wins, missing by only 0.0022. Tangent estimates shine when the number sits close to a perfect square — here 862 is just 21 above 841.
Finding √862 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 | 862 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.7241379310 | 29.3620689655 | 2 |
| 2 | 29.3620689655 | 29.3576042278 | 29.3598365967 | 7 |
| 3 | 29.3598365967 | 29.3598364269 | 29.3598365118 | 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 √862 = 29.3598365118 to every decimal shown.
√862 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √862 the pattern is [29; 2, 1, 3, 1, 1, 9, 4, 2, 2, 2, 1, 5, …] with the block of 40 terms after the semicolon repeating forever (only the first 12 of the 40 are shown). A pattern that never ends is one more proof that √862 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 | 3.6 × 10⁻¹ |
| 59/2 | 29.5000000000 | 1.4 × 10⁻¹ |
| 88/3 | 29.3333333333 | 2.7 × 10⁻² |
| 323/11 | 29.3636363636 | 3.8 × 10⁻³ |
| 411/14 | 29.3571428571 | 2.7 × 10⁻³ |
| 734/25 | 29.3600000000 | 1.6 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 862y² = 1. Its smallest solution in positive whole numbers is x = 358,022,566,147,312,125,503, y = 12,194,296,994,921,665,128 — 21 digits for x, even though 862 is small, which is what makes Pell’s equation famous.
√862 in geometry and everyday measurements
- 862 square feet is 80.1 m². Laid out as a square — a small house footprint or a lot — it is about 29.36 ft (29 ft 4 in) on a side.
- 862 is not a sum of two whole-number squares — the prime factor 431 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √862 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 18 × 23 box, because 3² + 18² + 23² = 862.
Square roots near √862 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √859 | √859 | 29.3087 | No |
| √860 | 2√215 | 29.3258 | No |
| √861 | √861 | 29.3428 | No |
| √862 | √862 | 29.3598 | No |
| √863 | √863 | 29.3769 | No |
| √864 | 12√6 | 29.3939 | No |
| √865 | √865 | 29.4109 | No |
- The cube root of 862 is about 9.517052.
- Squaring undoes the root: (√862)² = 862, while 862² = 743,044 — the number whose square root is 862.
Frequently asked questions
What is the square root of 862?
The square root of 862 is √862, about 29.3598365118. The negative root, −29.359837, also squares to 862.
Is the square root of 862 rational or irrational?
Irrational. 862 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 √862 be simplified?
No. 862 = 2 × 431 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √862 rounded to two decimal places?
√862 ≈ 29.36 to two decimal places (29.4 to one, 29.360 to three). Check: 29.36² = 862.0096, close to 862.