√422 at a glance
- Exact value
- √422
- Decimal (10 places)
- 20.5426385842
- Rounded
- 20.5 · 20.54 · 20.543
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.542639
- Prime factorization
- 2 × 211
- Cube root
- 7.500741
How to simplify √422
The prime factorization of 422 is 2 × 211. Every prime appears only once, so there is no pair to bring outside the radical — √422 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 422, 2 and 211 appear an odd number of times, so √422 is irrational and 20.5426385842 is a rounded value.
Where √422 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √422 lies between 20 and 21. 422 is 22 above 400 and 19 below 441, so the root is closer to 21.
- Straight line between 400 and 441: 20.5366 (0.03% low)
- Tangent from 20, i.e. 20 + 22 ÷ 40: 20.5500 (0.04% high)
- Tangent from 21, i.e. 21 − 19 ÷ 42: 20.5476 (0.02% high)
For √422 the tangent at 21 wins, missing by only 0.005. Tangent estimates shine when the number sits close to a perfect square — here 422 is just 19 below 441.
Finding √422 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, 21 (21² = 441):
| Step | Guess x | 422 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 20.0952380952 | 20.5476190476 | 2 |
| 2 | 20.5476190476 | 20.5376593279 | 20.5426391878 | 6 |
| 3 | 20.5426391878 | 20.5426379806 | 20.5426385842 | 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 √422 = 20.5426385842 to every decimal shown.
√422 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √422 the pattern is [20; 1, 1, 5, 2, 1, 3, 20, 3, 1, 2, 5, 1, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √422 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 |
|---|---|---|
| 20/1 | 20.0000000000 | 5.4 × 10⁻¹ |
| 21/1 | 21.0000000000 | 4.6 × 10⁻¹ |
| 41/2 | 20.5000000000 | 4.3 × 10⁻² |
| 226/11 | 20.5454545455 | 2.8 × 10⁻³ |
| 493/24 | 20.5416666667 | 9.7 × 10⁻⁴ |
| 719/35 | 20.5428571429 | 2.2 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 422y² = 1. Its smallest solution in positive whole numbers is x = 7,022,501, y = 341,850.
√422 in geometry and everyday measurements
- A square garage floor of 422 square feet measures about 20.54 ft (20 ft 7 in) per side, and its corner-to-corner diagonal is √844 ≈ 29.1 ft.
- 422 is not a sum of two whole-number squares — the prime factor 211 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √422 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 14 × 15 box, because 1² + 14² + 15² = 422.
Square roots near √422 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √419 | √419 | 20.4695 | No |
| √420 | 2√105 | 20.4939 | No |
| √421 | √421 | 20.5183 | No |
| √422 | √422 | 20.5426 | No |
| √423 | 3√47 | 20.5670 | No |
| √424 | 2√106 | 20.5913 | No |
| √425 | 5√17 | 20.6155 | No |
- The cube root of 422 is about 7.500741.
- Squaring undoes the root: (√422)² = 422, while 422² = 178,084 — the number whose square root is 422.
Frequently asked questions
What is the square root of 422?
The square root of 422 is √422, about 20.5426385842. The negative root, −20.542639, also squares to 422.
Is the square root of 422 rational or irrational?
Irrational. 422 is not a perfect square — it falls between 400 and 441 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √422 be simplified?
No. 422 = 2 × 211 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √422 rounded to two decimal places?
√422 ≈ 20.54 to two decimal places (20.5 to one, 20.543 to three). Check: 20.54² = 421.8916, close to 422.