√506 at a glance
- Exact value
- √506
- Decimal (10 places)
- 22.4944437584
- Rounded
- 22.5 · 22.49 · 22.494
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.494444
- Prime factorization
- 2 × 11 × 23
- Cube root
- 7.968627
How to simplify √506
The prime factorization of 506 is 2 × 11 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √506 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 506, 2, 11 and 23 appear an odd number of times, so √506 is irrational and 22.4944437584 is a rounded value.
Where √506 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √506 lies between 22 and 23. 506 is 22 above 484 and 23 below 529, so the root is closer to 22.
- Straight line between 484 and 529: 22.4889 (0.02% low)
- Tangent from 22, i.e. 22 + 22 ÷ 44: 22.5000 (0.02% high)
- Tangent from 23, i.e. 23 − 23 ÷ 46: 22.5000 (0.02% high)
For √506 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.
Finding √506 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 | 506 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 23.0000000000 | 22.5000000000 | 2 |
| 2 | 22.5000000000 | 22.4888888889 | 22.4944444444 | 6 |
| 3 | 22.4944444444 | 22.4944430724 | 22.4944437584 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √506 = 22.4944437584 to every decimal shown.
√506 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √506 the pattern is [22; 2, 44] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √506 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 |
|---|---|---|
| 22/1 | 22.0000000000 | 4.9 × 10⁻¹ |
| 45/2 | 22.5000000000 | 5.6 × 10⁻³ |
| 2,002/89 | 22.4943820225 | 6.2 × 10⁻⁵ |
| 4,049/180 | 22.4944444444 | 6.9 × 10⁻⁷ |
| 180,158/8,009 | 22.4944437508 | 7.6 × 10⁻⁹ |
| 364,365/16,198 | 22.4944437585 | 8.5 × 10⁻¹¹ |
The same fractions solve Pell’s equation, x² − 506y² = 1. Its smallest solution in positive whole numbers is x = 45, y = 2.
√506 in geometry and everyday measurements
- A square garage floor of 506 square feet measures about 22.49 ft (22 ft 6 in) per side, and its corner-to-corner diagonal is √1012 ≈ 31.8 ft.
- 506 is not a sum of two whole-number squares — the prime factor 11 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √506 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 8 × 21 box, because 1² + 8² + 21² = 506.
Square roots near √506 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √503 | √503 | 22.4277 | No |
| √504 | 6√14 | 22.4499 | No |
| √505 | √505 | 22.4722 | No |
| √506 | √506 | 22.4944 | No |
| √507 | 13√3 | 22.5167 | No |
| √508 | 2√127 | 22.5389 | No |
| √509 | √509 | 22.5610 | No |
- The cube root of 506 is about 7.968627.
- Squaring undoes the root: (√506)² = 506, while 506² = 256,036 — the number whose square root is 506.
Frequently asked questions
What is the square root of 506?
The square root of 506 is √506, about 22.4944437584. The negative root, −22.494444, also squares to 506.
Is the square root of 506 rational or irrational?
Irrational. 506 is not a perfect square — it falls between 484 and 529 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √506 be simplified?
No. 506 = 2 × 11 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √506 rounded to two decimal places?
√506 ≈ 22.49 to two decimal places (22.5 to one, 22.494 to three). Check: 22.49² = 505.8001, close to 506.