√822 at a glance
- Exact value
- √822
- Decimal (10 places)
- 28.6705423737
- Rounded
- 28.7 · 28.67 · 28.671
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.670542
- Prime factorization
- 2 × 3 × 137
- Cube root
- 9.367505
How to simplify √822
The prime factorization of 822 is 2 × 3 × 137. Every prime appears only once, so there is no pair to bring outside the radical — √822 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 822, 2, 3 and 137 appear an odd number of times, so √822 is irrational and 28.6705423737 is a rounded value.
Where √822 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √822 lies between 28 and 29. 822 is 38 above 784 and 19 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.6667 (0.01% low)
- Tangent from 28, i.e. 28 + 38 ÷ 56: 28.6786 (0.03% high)
- Tangent from 29, i.e. 29 − 19 ÷ 58: 28.6724 (0.01% high)
For √822 the tangent at 29 wins, missing by only 0.0019. Tangent estimates shine when the number sits close to a perfect square — here 822 is just 19 below 841.
Finding √822 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 | 822 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.3448275862 | 28.6724137931 | 2 |
| 2 | 28.6724137931 | 28.6686710764 | 28.6705424347 | 7 |
| 3 | 28.6705424347 | 28.6705423126 | 28.6705423737 | 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 √822 = 28.6705423737 to every decimal shown.
√822 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √822 the pattern is [28; 1, 2, 28, 2, 1, 56] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √822 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 | 6.7 × 10⁻¹ |
| 29/1 | 29.0000000000 | 3.3 × 10⁻¹ |
| 86/3 | 28.6666666667 | 3.9 × 10⁻³ |
| 2,437/85 | 28.6705882353 | 4.6 × 10⁻⁵ |
| 4,960/173 | 28.6705202312 | 2.2 × 10⁻⁵ |
| 7,397/258 | 28.6705426357 | 2.6 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 822y² = 1. Its smallest solution in positive whole numbers is x = 7,397, y = 258.
√822 in geometry and everyday measurements
- 822 square feet is 76.4 m². Laid out as a square — a small house footprint or a lot — it is about 28.67 ft (28 ft 8 in) on a side.
- 822 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √822 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 14 × 25 box, because 1² + 14² + 25² = 822.
Square roots near √822 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √819 | 3√91 | 28.6182 | No |
| √820 | 2√205 | 28.6356 | No |
| √821 | √821 | 28.6531 | No |
| √822 | √822 | 28.6705 | No |
| √823 | √823 | 28.6880 | No |
| √824 | 2√206 | 28.7054 | No |
| √825 | 5√33 | 28.7228 | No |
- The cube root of 822 is about 9.367505.
- Squaring undoes the root: (√822)² = 822, while 822² = 675,684 — the number whose square root is 822.
Frequently asked questions
What is the square root of 822?
The square root of 822 is √822, about 28.6705423737. The negative root, −28.670542, also squares to 822.
Is the square root of 822 rational or irrational?
Irrational. 822 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 √822 be simplified?
No. 822 = 2 × 3 × 137 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √822 rounded to two decimal places?
√822 ≈ 28.67 to two decimal places (28.7 to one, 28.671 to three). Check: 28.67² = 821.9689, close to 822.