√510 at a glance
- Exact value
- √510
- Decimal (10 places)
- 22.5831795813
- Rounded
- 22.6 · 22.58 · 22.583
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.583180
- Prime factorization
- 2 × 3 × 5 × 17
- Cube root
- 7.989570
How to simplify √510
The prime factorization of 510 is 2 × 3 × 5 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √510 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 510, 2, 3, 5 and 17 appear an odd number of times, so √510 is irrational and 22.5831795813 is a rounded value.
Where √510 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √510 lies between 22 and 23. 510 is 26 above 484 and 19 below 529, so the root is closer to 23.
- Straight line between 484 and 529: 22.5778 (0.02% low)
- Tangent from 22, i.e. 22 + 26 ÷ 44: 22.5909 (0.03% high)
- Tangent from 23, i.e. 23 − 19 ÷ 46: 22.5870 (0.02% high)
For √510 the tangent at 23 wins, missing by only 0.0038. Tangent estimates shine when the number sits close to a perfect square — here 510 is just 19 below 529.
Finding √510 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 | 510 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 22.1739130435 | 22.5869565217 | 2 |
| 2 | 22.5869565217 | 22.5794032724 | 22.5831798971 | 6 |
| 3 | 22.5831798971 | 22.5831792655 | 22.5831795813 | 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 √510 = 22.5831795813 to every decimal shown.
√510 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √510 the pattern is [22; 1, 1, 2, 1, 1, 44] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √510 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 | 5.8 × 10⁻¹ |
| 23/1 | 23.0000000000 | 4.2 × 10⁻¹ |
| 45/2 | 22.5000000000 | 8.3 × 10⁻² |
| 113/5 | 22.6000000000 | 1.7 × 10⁻² |
| 158/7 | 22.5714285714 | 1.2 × 10⁻² |
| 271/12 | 22.5833333333 | 1.5 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 510y² = 1. Its smallest solution in positive whole numbers is x = 271, y = 12.
√510 in geometry and everyday measurements
- A square garage floor of 510 square feet measures about 22.58 ft (22 ft 7 in) per side, and its corner-to-corner diagonal is √1020 ≈ 31.9 ft.
- 510 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 √510 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 22 box, because 1² + 5² + 22² = 510.
Square roots near √510 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √507 | 13√3 | 22.5167 | No |
| √508 | 2√127 | 22.5389 | No |
| √509 | √509 | 22.5610 | No |
| √510 | √510 | 22.5832 | No |
| √511 | √511 | 22.6053 | No |
| √512 | 16√2 | 22.6274 | No |
| √513 | 3√57 | 22.6495 | No |
- The cube root of 510 is about 7.989570.
- Squaring undoes the root: (√510)² = 510, while 510² = 260,100 — the number whose square root is 510.
Frequently asked questions
What is the square root of 510?
The square root of 510 is √510, about 22.5831795813. The negative root, −22.583180, also squares to 510.
Is the square root of 510 rational or irrational?
Irrational. 510 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 √510 be simplified?
No. 510 = 2 × 3 × 5 × 17 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √510 rounded to two decimal places?
√510 ≈ 22.58 to two decimal places (22.6 to one, 22.583 to three). Check: 22.58² = 509.8564, close to 510.