√482 at a glance
- Exact value
- √482
- Decimal (10 places)
- 21.9544984001
- Rounded
- 22.0 · 21.95 · 21.954
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.954498
- Prime factorization
- 2 × 241
- Cube root
- 7.840595
How to simplify √482
The prime factorization of 482 is 2 × 241. Every prime appears only once, so there is no pair to bring outside the radical — √482 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 482, 2 and 241 appear an odd number of times, so √482 is irrational and 21.9544984001 is a rounded value.
Where √482 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √482 lies between 21 and 22. 482 is 41 above 441 and 2 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.9535 (0% low)
- Tangent from 21, i.e. 21 + 41 ÷ 42: 21.9762 (0.1% high)
- Tangent from 22, i.e. 22 − 2 ÷ 44: 21.9545 (0% high)
For √482 the tangent at 22 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 482 is just 2 below 484.
Finding √482 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, 22 (22² = 484):
| Step | Guess x | 482 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.9090909091 | 21.9545454545 | 4 |
| 2 | 21.9545454545 | 21.9544513458 | 21.9544984002 | 10 |
| 3 | 21.9544984002 | 21.9544984000 | 21.9544984001 | all 10 shown |
The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √482 = 21.9544984001 to every decimal shown.
√482 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √482 the pattern is [21; 1, 20, 1, 42] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √482 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 |
|---|---|---|
| 21/1 | 21.0000000000 | 9.5 × 10⁻¹ |
| 22/1 | 22.0000000000 | 4.6 × 10⁻² |
| 461/21 | 21.9523809524 | 2.1 × 10⁻³ |
| 483/22 | 21.9545454545 | 4.7 × 10⁻⁵ |
| 20,747/945 | 21.9544973545 | 1.0 × 10⁻⁶ |
| 21,230/967 | 21.9544984488 | 4.9 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 482y² = 1. Its smallest solution in positive whole numbers is x = 483, y = 22.
√482 in geometry and everyday measurements
- A square garage floor of 482 square feet measures about 21.95 ft (21 ft 11 in) per side, and its corner-to-corner diagonal is √964 ≈ 31 ft.
- 482 = 11² + 19², so by the Pythagorean theorem √482 is the diagonal of a 11 × 19 rectangle — and the distance between the points (0, 0) and (11, 19) on a grid.
Square roots near √482 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √479 | √479 | 21.8861 | No |
| √480 | 4√30 | 21.9089 | No |
| √481 | √481 | 21.9317 | No |
| √482 | √482 | 21.9545 | No |
| √483 | √483 | 21.9773 | No |
| √484 | 22 | 22.0000 | Yes |
| √485 | √485 | 22.0227 | No |
- The cube root of 482 is about 7.840595.
- Squaring undoes the root: (√482)² = 482, while 482² = 232,324 — the number whose square root is 482.
Frequently asked questions
What is the square root of 482?
The square root of 482 is √482, about 21.9544984001. The negative root, −21.954498, also squares to 482.
Is the square root of 482 rational or irrational?
Irrational. 482 is not a perfect square — it falls between 441 and 484 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √482 be simplified?
No. 482 = 2 × 241 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √482 rounded to two decimal places?
√482 ≈ 21.95 to two decimal places (22.0 to one, 21.954 to three). Check: 21.95² = 481.8025, close to 482.