√830 at a glance
- Exact value
- √830
- Decimal (10 places)
- 28.8097205818
- Rounded
- 28.8 · 28.81 · 28.810
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.809721
- Prime factorization
- 2 × 5 × 83
- Cube root
- 9.397796
How to simplify √830
The prime factorization of 830 is 2 × 5 × 83. Every prime appears only once, so there is no pair to bring outside the radical — √830 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 830, 2, 5 and 83 appear an odd number of times, so √830 is irrational and 28.8097205818 is a rounded value.
Where √830 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √830 lies between 28 and 29. 830 is 46 above 784 and 11 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.8070 (0.01% low)
- Tangent from 28, i.e. 28 + 46 ÷ 56: 28.8214 (0.04% high)
- Tangent from 29, i.e. 29 − 11 ÷ 58: 28.8103 (0% high)
For √830 the tangent at 29 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 830 is just 11 below 841.
Finding √830 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 | 830 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.6206896552 | 28.8103448276 | 3 |
| 2 | 28.8103448276 | 28.8090963495 | 28.8097205885 | 8 |
| 3 | 28.8097205885 | 28.8097205750 | 28.8097205818 | 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 √830 = 28.8097205818 to every decimal shown.
√830 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √830 the pattern is [28; 1, 4, 3, 1, 10, 1, 3, 4, 1, 56] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √830 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 | 8.1 × 10⁻¹ |
| 29/1 | 29.0000000000 | 1.9 × 10⁻¹ |
| 144/5 | 28.8000000000 | 9.7 × 10⁻³ |
| 461/16 | 28.8125000000 | 2.8 × 10⁻³ |
| 605/21 | 28.8095238095 | 2.0 × 10⁻⁴ |
| 6,511/226 | 28.8097345133 | 1.4 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 830y² = 1. Its smallest solution in positive whole numbers is x = 146,411, y = 5,082.
√830 in geometry and everyday measurements
- 830 square feet is 77.1 m². Laid out as a square — a small house footprint or a lot — it is about 28.81 ft (28 ft 10 in) on a side.
- 830 is not a sum of two whole-number squares — the prime factor 83 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √830 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 10 × 27 box, because 1² + 10² + 27² = 830.
Square roots near √830 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √827 | √827 | 28.7576 | No |
| √828 | 6√23 | 28.7750 | No |
| √829 | √829 | 28.7924 | No |
| √830 | √830 | 28.8097 | No |
| √831 | √831 | 28.8271 | No |
| √832 | 8√13 | 28.8444 | No |
| √833 | 7√17 | 28.8617 | No |
- The cube root of 830 is about 9.397796.
- Squaring undoes the root: (√830)² = 830, while 830² = 688,900 — the number whose square root is 830.
Frequently asked questions
What is the square root of 830?
The square root of 830 is √830, about 28.8097205818. The negative root, −28.809721, also squares to 830.
Is the square root of 830 rational or irrational?
Irrational. 830 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 √830 be simplified?
No. 830 = 2 × 5 × 83 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √830 rounded to two decimal places?
√830 ≈ 28.81 to two decimal places (28.8 to one, 28.810 to three). Check: 28.81² = 830.0161, close to 830.