√522 at a glance
- Exact value
- 3√58
- Decimal (10 places)
- 22.8473193176
- Rounded
- 22.8 · 22.85 · 22.847
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.847319
- Prime factorization
- 2 × 3² × 29
- Cube root
- 8.051748
How to simplify √522
Look for the largest perfect square that divides 522. Here it is 9 (3²), because 522 = 9 × 58 and 58 has no square factor left:
The prime factorization tells the same story: 522 = 2 × 3² × 29. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 29 stays inside.
Check: (3√58)² = 3² × 58 = 9 × 58 = 522. As a decimal, 3√58 = 3 × 7.6157731059 ≈ 22.8473193176.
Where √522 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √522 lies between 22 and 23. 522 is 38 above 484 and 7 below 529, so the root is closer to 23.
- Straight line between 484 and 529: 22.8444 (0.01% low)
- Tangent from 22, i.e. 22 + 38 ÷ 44: 22.8636 (0.07% high)
- Tangent from 23, i.e. 23 − 7 ÷ 46: 22.8478 (0% high)
For √522 the tangent at 23 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 522 is just 7 below 529.
Finding √522 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, 23 (23² = 529):
| Step | Guess x | 522 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 22.6956521739 | 22.8478260870 | 3 |
| 2 | 22.8478260870 | 22.8468125595 | 22.8473193232 | 8 |
| 3 | 22.8473193232 | 22.8473193120 | 22.8473193176 | 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 √522 = 22.8473193176 to every decimal shown.
√522 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √522 the pattern is [22; 1, 5, 1, 1, 4, 1, 1, 5, 1, 44] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √522 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 | 8.5 × 10⁻¹ |
| 23/1 | 23.0000000000 | 1.5 × 10⁻¹ |
| 137/6 | 22.8333333333 | 1.4 × 10⁻² |
| 160/7 | 22.8571428571 | 9.8 × 10⁻³ |
| 297/13 | 22.8461538462 | 1.2 × 10⁻³ |
| 1,348/59 | 22.8474576271 | 1.4 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 522y² = 1. Its smallest solution in positive whole numbers is x = 19,603, y = 858.
√522 in geometry and everyday measurements
- A square garage floor of 522 square feet measures about 22.85 ft (22 ft 10 in) per side, and its corner-to-corner diagonal is √1044 ≈ 32.3 ft.
- 522 = 9² + 21², so by the Pythagorean theorem √522 is the diagonal of a 9 × 21 rectangle — and the distance between the points (0, 0) and (9, 21) on a grid.
- Since √522 = 3√58, a length of √522 is exactly 3 copies of the length √58 laid end to end.
Square roots near √522 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √519 | √519 | 22.7816 | No |
| √520 | 2√130 | 22.8035 | No |
| √521 | √521 | 22.8254 | No |
| √522 | 3√58 | 22.8473 | No |
| √523 | √523 | 22.8692 | No |
| √524 | 2√131 | 22.8910 | No |
| √525 | 5√21 | 22.9129 | No |
- The cube root of 522 is about 8.051748.
- Squaring undoes the root: (√522)² = 522, while 522² = 272,484 — the number whose square root is 522.
Frequently asked questions
What is the square root of 522?
The square root of 522 is 3√58 in simplest radical form, which is about 22.8473193176. The negative root, −22.847319, also squares to 522.
Is the square root of 522 rational or irrational?
Irrational. 522 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 √522 be simplified?
Yes. The largest perfect square dividing 522 is 9, so √522 = √9 × √58 = 3√58.
What is √522 rounded to two decimal places?
√522 ≈ 22.85 to two decimal places (22.8 to one, 22.847 to three). Check: 22.85² = 522.1225, close to 522.