√520 at a glance
- Exact value
- 2√130
- Decimal (10 places)
- 22.8035085020
- Rounded
- 22.8 · 22.80 · 22.804
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.803509
- Prime factorization
- 2³ × 5 × 13
- Cube root
- 8.041452
How to simplify √520
Look for the largest perfect square that divides 520. Here it is 4 (2²), because 520 = 4 × 130 and 130 has no square factor left:
The prime factorization tells the same story: 520 = 2³ × 5 × 13. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 5 × 13 stays inside.
Check: (2√130)² = 2² × 130 = 4 × 130 = 520. As a decimal, 2√130 = 2 × 11.401754251 ≈ 22.8035085020.
Where √520 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √520 lies between 22 and 23. 520 is 36 above 484 and 9 below 529, so the root is closer to 23.
- Straight line between 484 and 529: 22.8000 (0.02% low)
- Tangent from 22, i.e. 22 + 36 ÷ 44: 22.8182 (0.06% high)
- Tangent from 23, i.e. 23 − 9 ÷ 46: 22.8043 (0% high)
For √520 the tangent at 23 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 520 is just 9 below 529.
Finding √520 with the Babylonian method
Picture a rectangle with an area of 520 and one side x; the other side must be 520 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √520.
Start from the nearest whole number, 23 (23² = 529):
| Step | Guess x | 520 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 22.6086956522 | 22.8043478261 | 3 |
| 2 | 22.8043478261 | 22.8026692088 | 22.8035085174 | 7 |
| 3 | 22.8035085174 | 22.8035084865 | 22.8035085020 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √520 = 22.8035085020 to every decimal shown.
√520 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √520 the pattern is [22; 1, 4, 11, 4, 1, 44] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √520 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 | 8.0 × 10⁻¹ |
| 23/1 | 23.0000000000 | 2.0 × 10⁻¹ |
| 114/5 | 22.8000000000 | 3.5 × 10⁻³ |
| 1,277/56 | 22.8035714286 | 6.3 × 10⁻⁵ |
| 5,222/229 | 22.8034934498 | 1.5 × 10⁻⁵ |
| 6,499/285 | 22.8035087719 | 2.7 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 520y² = 1. Its smallest solution in positive whole numbers is x = 6,499, y = 285.
√520 in geometry and everyday measurements
- A square garage floor of 520 square feet measures about 22.8 ft (22 ft 10 in) per side, and its corner-to-corner diagonal is √1040 ≈ 32.2 ft.
- 520 = 6² + 22² = 14² + 18², so by the Pythagorean theorem √520 is the diagonal of rectangles measuring 6 × 22 and 14 × 18 — and the distance between the points (0, 0) and (6, 22) on a grid.
- Since √520 = 2√130, a length of √520 is exactly 2 copies of the length √130 laid end to end.
Square roots near √520 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √522 | 3√58 | 22.8473 | No |
| √523 | √523 | 22.8692 | No |
- The cube root of 520 is about 8.041452.
- Because 520 = 4 × 130, the root is twice √130: 2 × 11.401754 ≈ 22.803509.
Frequently asked questions
What is the square root of 520?
The square root of 520 is 2√130 in simplest radical form, which is about 22.8035085020. The negative root, −22.803509, also squares to 520.
Is the square root of 520 rational or irrational?
Irrational. 520 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 √520 be simplified?
Yes. The largest perfect square dividing 520 is 4, so √520 = √4 × √130 = 2√130.
What is √520 rounded to two decimal places?
√520 ≈ 22.80 to two decimal places (22.8 to one, 22.804 to three). Check: 22.80² = 519.84, close to 520.