√426 at a glance
- Exact value
- √426
- Decimal (10 places)
- 20.6397674406
- Rounded
- 20.6 · 20.64 · 20.640
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.639767
- Prime factorization
- 2 × 3 × 71
- Cube root
- 7.524365
How to simplify √426
The prime factorization of 426 is 2 × 3 × 71. Every prime appears only once, so there is no pair to bring outside the radical — √426 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 426, 2, 3 and 71 appear an odd number of times, so √426 is irrational and 20.6397674406 is a rounded value.
Where √426 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √426 lies between 20 and 21. 426 is 26 above 400 and 15 below 441, so the root is closer to 21.
- Straight line between 400 and 441: 20.6341 (0.03% low)
- Tangent from 20, i.e. 20 + 26 ÷ 40: 20.6500 (0.05% high)
- Tangent from 21, i.e. 21 − 15 ÷ 42: 20.6429 (0.01% high)
For √426 the tangent at 21 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 426 is just 15 below 441.
Finding √426 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 | 426 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 20.2857142857 | 20.6428571429 | 2 |
| 2 | 20.6428571429 | 20.6366782007 | 20.6397676718 | 6 |
| 3 | 20.6397676718 | 20.6397672093 | 20.6397674406 | 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 √426 = 20.6397674406 to every decimal shown.
√426 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √426 the pattern is [20; 1, 1, 1, 3, 2, 6, 2, 3, 1, 1, 1, 40] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √426 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 | 6.4 × 10⁻¹ |
| 21/1 | 21.0000000000 | 3.6 × 10⁻¹ |
| 41/2 | 20.5000000000 | 1.4 × 10⁻¹ |
| 62/3 | 20.6666666667 | 2.7 × 10⁻² |
| 227/11 | 20.6363636364 | 3.4 × 10⁻³ |
| 516/25 | 20.6400000000 | 2.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 426y² = 1. Its smallest solution in positive whole numbers is x = 88,751, y = 4,300.
√426 in geometry and everyday measurements
- A square garage floor of 426 square feet measures about 20.64 ft (20 ft 8 in) per side, and its corner-to-corner diagonal is √852 ≈ 29.2 ft.
- 426 is not a sum of two whole-number squares — the prime factor 3 and 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √426 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 20 box, because 1² + 5² + 20² = 426.
Square roots near √426 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √423 | 3√47 | 20.5670 | No |
| √424 | 2√106 | 20.5913 | No |
| √425 | 5√17 | 20.6155 | No |
| √426 | √426 | 20.6398 | No |
| √427 | √427 | 20.6640 | No |
| √428 | 2√107 | 20.6882 | No |
| √429 | √429 | 20.7123 | No |
- The cube root of 426 is about 7.524365.
- Squaring undoes the root: (√426)² = 426, while 426² = 181,476 — the number whose square root is 426.
Frequently asked questions
What is the square root of 426?
The square root of 426 is √426, about 20.6397674406. The negative root, −20.639767, also squares to 426.
Is the square root of 426 rational or irrational?
Irrational. 426 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 √426 be simplified?
No. 426 = 2 × 3 × 71 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √426 rounded to two decimal places?
√426 ≈ 20.64 to two decimal places (20.6 to one, 20.640 to three). Check: 20.64² = 426.0096, close to 426.