√406 at a glance
- Exact value
- √406
- Decimal (10 places)
- 20.1494416796
- Rounded
- 20.1 · 20.15 · 20.149
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.149442
- Prime factorization
- 2 × 7 × 29
- Cube root
- 7.404721
How to simplify √406
The prime factorization of 406 is 2 × 7 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √406 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 406, 2, 7 and 29 appear an odd number of times, so √406 is irrational and 20.1494416796 is a rounded value.
Where √406 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √406 lies between 20 and 21. 406 is 6 above 400 and 35 below 441, so the root is closer to 20.
- Straight line between 400 and 441: 20.1463 (0.02% low)
- Tangent from 20, i.e. 20 + 6 ÷ 40: 20.1500 (0% high)
- Tangent from 21, i.e. 21 − 35 ÷ 42: 20.1667 (0.09% high)
For √406 the tangent at 20 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 406 is just 6 above 400.
Finding √406 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, 20 (20² = 400):
| Step | Guess x | 406 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 20.3000000000 | 20.1500000000 | 3 |
| 2 | 20.1500000000 | 20.1488833747 | 20.1494416873 | 8 |
| 3 | 20.1494416873 | 20.1494416719 | 20.1494416796 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √406 = 20.1494416796 to every decimal shown.
√406 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √406 the pattern is [20; 6, 1, 2, 4, 7, 1, 4, 1, 7, 4, 2, 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 √406 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 | 1.5 × 10⁻¹ |
| 121/6 | 20.1666666667 | 1.7 × 10⁻² |
| 141/7 | 20.1428571429 | 6.6 × 10⁻³ |
| 403/20 | 20.1500000000 | 5.6 × 10⁻⁴ |
| 1,753/87 | 20.1494252874 | 1.6 × 10⁻⁵ |
| 12,674/629 | 20.1494435612 | 1.9 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 406y² = 1. Its smallest solution in positive whole numbers is x = 59,468,095, y = 2,951,352.
√406 in geometry and everyday measurements
- A square garage floor of 406 square feet measures about 20.15 ft (20 ft 2 in) per side, and its corner-to-corner diagonal is √812 ≈ 28.5 ft.
- 406 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √406 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 18 box, because 1² + 9² + 18² = 406.
Square roots near √406 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √403 | √403 | 20.0749 | No |
| √404 | 2√101 | 20.0998 | No |
| √405 | 9√5 | 20.1246 | No |
| √406 | √406 | 20.1494 | No |
| √407 | √407 | 20.1742 | No |
| √408 | 2√102 | 20.1990 | No |
| √409 | √409 | 20.2237 | No |
- The cube root of 406 is about 7.404721.
- Squaring undoes the root: (√406)² = 406, while 406² = 164,836 — the number whose square root is 406.
Frequently asked questions
What is the square root of 406?
The square root of 406 is √406, about 20.1494416796. The negative root, −20.149442, also squares to 406.
Is the square root of 406 rational or irrational?
Irrational. 406 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 √406 be simplified?
No. 406 = 2 × 7 × 29 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √406 rounded to two decimal places?
√406 ≈ 20.15 to two decimal places (20.1 to one, 20.149 to three). Check: 20.15² = 406.0225, close to 406.