√518 at a glance
- Exact value
- √518
- Decimal (10 places)
- 22.7596133535
- Rounded
- 22.8 · 22.76 · 22.760
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.759613
- Prime factorization
- 2 × 7 × 37
- Cube root
- 8.031129
How to simplify √518
The prime factorization of 518 is 2 × 7 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √518 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 518, 2, 7 and 37 appear an odd number of times, so √518 is irrational and 22.7596133535 is a rounded value.
Where √518 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √518 lies between 22 and 23. 518 is 34 above 484 and 11 below 529, so the root is closer to 23.
- Straight line between 484 and 529: 22.7556 (0.02% low)
- Tangent from 22, i.e. 22 + 34 ÷ 44: 22.7727 (0.06% high)
- Tangent from 23, i.e. 23 − 11 ÷ 46: 22.7609 (0.01% high)
For √518 the tangent at 23 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 518 is just 11 below 529.
Finding √518 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 | 518 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 22.5217391304 | 22.7608695652 | 2 |
| 2 | 22.7608695652 | 22.7583572111 | 22.7596133881 | 7 |
| 3 | 22.7596133881 | 22.7596133188 | 22.7596133535 | 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 √518 = 22.7596133535 to every decimal shown.
√518 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √518 the pattern is [22; 1, 3, 6, 3, 1, 44] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √518 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 | 7.6 × 10⁻¹ |
| 23/1 | 23.0000000000 | 2.4 × 10⁻¹ |
| 91/4 | 22.7500000000 | 9.6 × 10⁻³ |
| 569/25 | 22.7600000000 | 3.9 × 10⁻⁴ |
| 1,798/79 | 22.7594936709 | 1.2 × 10⁻⁴ |
| 2,367/104 | 22.7596153846 | 2.0 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 518y² = 1. Its smallest solution in positive whole numbers is x = 2,367, y = 104.
√518 in geometry and everyday measurements
- A square garage floor of 518 square feet measures about 22.76 ft (22 ft 9 in) per side, and its corner-to-corner diagonal is √1036 ≈ 32.2 ft.
- 518 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √518 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 15 × 17 box, because 2² + 15² + 17² = 518.
Square roots near √518 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √515 | √515 | 22.6936 | No |
| √516 | 2√129 | 22.7156 | No |
| √517 | √517 | 22.7376 | No |
| √518 | √518 | 22.7596 | No |
| √519 | √519 | 22.7816 | No |
| √520 | 2√130 | 22.8035 | No |
| √521 | √521 | 22.8254 | No |
- The cube root of 518 is about 8.031129.
- Squaring undoes the root: (√518)² = 518, while 518² = 268,324 — the number whose square root is 518.
Frequently asked questions
What is the square root of 518?
The square root of 518 is √518, about 22.7596133535. The negative root, −22.759613, also squares to 518.
Is the square root of 518 rational or irrational?
Irrational. 518 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 √518 be simplified?
No. 518 = 2 × 7 × 37 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √518 rounded to two decimal places?
√518 ≈ 22.76 to two decimal places (22.8 to one, 22.760 to three). Check: 22.76² = 518.0176, close to 518.