√505 at a glance
- Exact value
- √505
- Decimal (10 places)
- 22.4722050542
- Rounded
- 22.5 · 22.47 · 22.472
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.472205
- Prime factorization
- 5 × 101
- Cube root
- 7.963374
How to simplify √505
The prime factorization of 505 is 5 × 101. Every prime appears only once, so there is no pair to bring outside the radical — √505 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 505, 5 and 101 appear an odd number of times, so √505 is irrational and 22.4722050542 is a rounded value.
Where √505 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √505 lies between 22 and 23. 505 is 21 above 484 and 24 below 529, so the root is closer to 22.
- Straight line between 484 and 529: 22.4667 (0.02% low)
- Tangent from 22, i.e. 22 + 21 ÷ 44: 22.4773 (0.02% high)
- Tangent from 23, i.e. 23 − 24 ÷ 46: 22.4783 (0.03% high)
For √505 the tangent at 22 wins, missing by only 0.0051. Tangent estimates shine when the number sits close to a perfect square — here 505 is just 21 above 484.
Finding √505 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 505: following the tangent line down to zero simplifies to averaging x with 505 ÷ x.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 505 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 22.9545454545 | 22.4772727273 | 2 |
| 2 | 22.4772727273 | 22.4671385238 | 22.4722056255 | 6 |
| 3 | 22.4722056255 | 22.4722044830 | 22.4722050542 | 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 √505 = 22.4722050542 to every decimal shown.
√505 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √505 the pattern is [22; 2, 8, 2, 44] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √505 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 | 4.7 × 10⁻¹ |
| 45/2 | 22.5000000000 | 2.8 × 10⁻² |
| 382/17 | 22.4705882353 | 1.6 × 10⁻³ |
| 809/36 | 22.4722222222 | 1.7 × 10⁻⁵ |
| 35,978/1,601 | 22.4722048720 | 1.8 × 10⁻⁷ |
| 72,765/3,238 | 22.4722050649 | 1.1 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 505y² = 1. Its smallest solution in positive whole numbers is x = 809, y = 36.
√505 in geometry and everyday measurements
- A square garage floor of 505 square feet measures about 22.47 ft (22 ft 6 in) per side, and its corner-to-corner diagonal is √1010 ≈ 31.8 ft.
- 505 = 8² + 21² = 12² + 19², so by the Pythagorean theorem √505 is the diagonal of rectangles measuring 8 × 21 and 12 × 19 — and the distance between the points (0, 0) and (8, 21) on a grid.
Square roots near √505 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √502 | √502 | 22.4054 | No |
| √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 |
- The cube root of 505 is about 7.963374.
- Squaring undoes the root: (√505)² = 505, while 505² = 255,025 — the number whose square root is 505.
Frequently asked questions
What is the square root of 505?
The square root of 505 is √505, about 22.4722050542. The negative root, −22.472205, also squares to 505.
Is the square root of 505 rational or irrational?
Irrational. 505 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 √505 be simplified?
No. 505 = 5 × 101 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √505 rounded to two decimal places?
√505 ≈ 22.47 to two decimal places (22.5 to one, 22.472 to three). Check: 22.47² = 504.9009, close to 505.