√22 at a glance
- Exact value
- √22
- Decimal (10 places)
- 4.6904157598
- Rounded
- 4.7 · 4.69 · 4.690
- Perfect square?
- No — between 4² and 5²
- Rational?
- Irrational
- Both square roots
- ±4.690416
- Prime factorization
- 2 × 11
- Cube root
- 2.802039
How to simplify √22
The prime factorization of 22 is 2 × 11. Every prime appears only once, so there is no pair to bring outside the radical — √22 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 22, 2 and 11 appear an odd number of times, so √22 is irrational and 4.6904157598 is a rounded value.
Where √22 sits between perfect squares
16 = 4² and 25 = 5² are the nearest perfect squares, so √22 lies between 4 and 5. 22 is 6 above 16 and 3 below 25, so the root is closer to 5.
- Straight line between 16 and 25: 4.6667 (0.51% low)
- Tangent from 4, i.e. 4 + 6 ÷ 8: 4.7500 (1.27% high)
- Tangent from 5, i.e. 5 − 3 ÷ 10: 4.7000 (0.2% high)
For √22 the tangent at 5 wins, missing by only 0.0096. Tangent estimates shine when the number sits close to a perfect square — here 22 is just 3 below 25.
Finding √22 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, 5 (5² = 25):
| Step | Guess x | 22 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 5.0000000000 | 4.4000000000 | 4.7000000000 | 2 |
| 2 | 4.7000000000 | 4.6808510638 | 4.6904255319 | 5 |
| 3 | 4.6904255319 | 4.6904059878 | 4.6904157598 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √22 = 4.6904157598 to every decimal shown.
√22 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √22 the pattern is [4; 1, 2, 4, 2, 1, 8] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √22 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 |
|---|---|---|
| 4/1 | 4.0000000000 | 6.9 × 10⁻¹ |
| 5/1 | 5.0000000000 | 3.1 × 10⁻¹ |
| 14/3 | 4.6666666667 | 2.4 × 10⁻² |
| 61/13 | 4.6923076923 | 1.9 × 10⁻³ |
| 136/29 | 4.6896551724 | 7.6 × 10⁻⁴ |
| 197/42 | 4.6904761905 | 6.0 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 22y² = 1. Its smallest solution in positive whole numbers is x = 197, y = 42.
√22 in geometry and everyday measurements
- A square tile with an area of 22 square inches has sides about 4.69 in long.
- 22 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √22 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 3 × 3 box, because 2² + 3² + 3² = 22.
Square roots near √22 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √19 | √19 | 4.3589 | No |
| √20 | 2√5 | 4.4721 | No |
| √21 | √21 | 4.5826 | No |
| √22 | √22 | 4.6904 | No |
| √23 | √23 | 4.7958 | No |
| √24 | 2√6 | 4.8990 | No |
| √25 | 5 | 5.0000 | Yes |
- The cube root of 22 is about 2.802039.
- Four times the radicand doubles the root: √88 = 2 × √22 ≈ 9.380832.
Frequently asked questions
What is the square root of 22?
The square root of 22 is √22, about 4.6904157598. The negative root, −4.690416, also squares to 22.
Is the square root of 22 rational or irrational?
Irrational. 22 is not a perfect square — it falls between 16 and 25 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √22 be simplified?
No. 22 = 2 × 11 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √22 rounded to two decimal places?
√22 ≈ 4.69 to two decimal places (4.7 to one, 4.690 to three). Check: 4.69² = 21.9961, close to 22.