√834 at a glance
- Exact value
- √834
- Decimal (10 places)
- 28.8790581564
- Rounded
- 28.9 · 28.88 · 28.879
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.879058
- Prime factorization
- 2 × 3 × 139
- Cube root
- 9.412869
How to simplify √834
The prime factorization of 834 is 2 × 3 × 139. Every prime appears only once, so there is no pair to bring outside the radical — √834 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 834, 2, 3 and 139 appear an odd number of times, so √834 is irrational and 28.8790581564 is a rounded value.
Where √834 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √834 lies between 28 and 29. 834 is 50 above 784 and 7 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.8772 (0.01% low)
- Tangent from 28, i.e. 28 + 50 ÷ 56: 28.8929 (0.05% high)
- Tangent from 29, i.e. 29 − 7 ÷ 58: 28.8793 (0% high)
For √834 the tangent at 29 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 834 is just 7 below 841.
Finding √834 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 | 834 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.7586206897 | 28.8793103448 | 3 |
| 2 | 28.8793103448 | 28.8788059701 | 28.8790581575 | 8 |
| 3 | 28.8790581575 | 28.8790581553 | 28.8790581564 | 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 √834 = 28.8790581564 to every decimal shown.
√834 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √834 the pattern is [28; 1, 7, 3, 1, 2, 1, 1, 1, 3, 2, 28, 2, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √834 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 |
|---|---|---|
| 28/1 | 28.0000000000 | 8.8 × 10⁻¹ |
| 29/1 | 29.0000000000 | 1.2 × 10⁻¹ |
| 231/8 | 28.8750000000 | 4.1 × 10⁻³ |
| 722/25 | 28.8800000000 | 9.4 × 10⁻⁴ |
| 953/33 | 28.8787878788 | 2.7 × 10⁻⁴ |
| 2,628/91 | 28.8791208791 | 6.3 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 834y² = 1. Its smallest solution in positive whole numbers is x = 6,552,578,705, y = 226,897,244.
√834 in geometry and everyday measurements
- 834 square feet is 77.5 m². Laid out as a square — a small house footprint or a lot — it is about 28.88 ft (28 ft 11 in) on a side.
- 834 is not a sum of two whole-number squares — the prime factor 3 and 139 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √834 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 28 box, because 1² + 7² + 28² = 834.
Square roots near √834 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √831 | √831 | 28.8271 | No |
| √832 | 8√13 | 28.8444 | No |
| √833 | 7√17 | 28.8617 | No |
| √834 | √834 | 28.8791 | No |
| √835 | √835 | 28.8964 | No |
| √836 | 2√209 | 28.9137 | No |
| √837 | 3√93 | 28.9310 | No |
- The cube root of 834 is about 9.412869.
- Squaring undoes the root: (√834)² = 834, while 834² = 695,556 — the number whose square root is 834.
Frequently asked questions
What is the square root of 834?
The square root of 834 is √834, about 28.8790581564. The negative root, −28.879058, also squares to 834.
Is the square root of 834 rational or irrational?
Irrational. 834 is not a perfect square — it falls between 784 and 841 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √834 be simplified?
No. 834 = 2 × 3 × 139 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √834 rounded to two decimal places?
√834 ≈ 28.88 to two decimal places (28.9 to one, 28.879 to three). Check: 28.88² = 834.0544, close to 834.