√622 at a glance
- Exact value
- √622
- Decimal (10 places)
- 24.9399278267
- Rounded
- 24.9 · 24.94 · 24.940
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.939928
- Prime factorization
- 2 × 311
- Cube root
- 8.536178
How to simplify √622
The prime factorization of 622 is 2 × 311. Every prime appears only once, so there is no pair to bring outside the radical — √622 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 622, 2 and 311 appear an odd number of times, so √622 is irrational and 24.9399278267 is a rounded value.
Where √622 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √622 lies between 24 and 25. 622 is 46 above 576 and 3 below 625, so the root is closer to 25.
- Straight line between 576 and 625: 24.9388 (0% low)
- Tangent from 24, i.e. 24 + 46 ÷ 48: 24.9583 (0.07% high)
- Tangent from 25, i.e. 25 − 3 ÷ 50: 24.9400 (0% high)
For √622 the tangent at 25 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 622 is just 3 below 625.
Finding √622 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, 25 (25² = 625):
| Step | Guess x | 622 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 24.8800000000 | 24.9400000000 | 4 |
| 2 | 24.9400000000 | 24.9398556536 | 24.9399278268 | 9 |
| 3 | 24.9399278268 | 24.9399278266 | 24.9399278267 | all 10 shown |
The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √622 = 24.9399278267 to every decimal shown.
√622 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √622 the pattern is [24; 1, 15, 1, 1, 1, 4, 1, 7, 2, 24, 2, 7, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √622 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 |
|---|---|---|
| 24/1 | 24.0000000000 | 9.4 × 10⁻¹ |
| 25/1 | 25.0000000000 | 6.0 × 10⁻² |
| 399/16 | 24.9375000000 | 2.4 × 10⁻³ |
| 424/17 | 24.9411764706 | 1.2 × 10⁻³ |
| 823/33 | 24.9393939394 | 5.3 × 10⁻⁴ |
| 1,247/50 | 24.9400000000 | 7.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 622y² = 1. Its smallest solution in positive whole numbers is x = 13,804,370,063, y = 553,504,812.
√622 in geometry and everyday measurements
- A square garage floor of 622 square feet measures about 24.94 ft (24 ft 11 in) per side, and its corner-to-corner diagonal is √1244 ≈ 35.3 ft.
- 622 is not a sum of two whole-number squares — the prime factor 311 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √622 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 17 × 18 box, because 3² + 17² + 18² = 622.
Square roots near √622 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √619 | √619 | 24.8797 | No |
| √620 | 2√155 | 24.8998 | No |
| √621 | 3√69 | 24.9199 | No |
| √622 | √622 | 24.9399 | No |
| √623 | √623 | 24.9600 | No |
| √624 | 4√39 | 24.9800 | No |
| √625 | 25 | 25.0000 | Yes |
- The cube root of 622 is about 8.536178.
- Squaring undoes the root: (√622)² = 622, while 622² = 386,884 — the number whose square root is 622.
Frequently asked questions
What is the square root of 622?
The square root of 622 is √622, about 24.9399278267. The negative root, −24.939928, also squares to 622.
Is the square root of 622 rational or irrational?
Irrational. 622 is not a perfect square — it falls between 576 and 625 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √622 be simplified?
No. 622 = 2 × 311 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √622 rounded to two decimal places?
√622 ≈ 24.94 to two decimal places (24.9 to one, 24.940 to three). Check: 24.94² = 622.0036, close to 622.