√446 at a glance
- Exact value
- √446
- Decimal (10 places)
- 21.1187120819
- Rounded
- 21.1 · 21.12 · 21.119
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.118712
- Prime factorization
- 2 × 223
- Cube root
- 7.640321
How to simplify √446
The prime factorization of 446 is 2 × 223. Every prime appears only once, so there is no pair to bring outside the radical — √446 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 446, 2 and 223 appear an odd number of times, so √446 is irrational and 21.1187120819 is a rounded value.
Where √446 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √446 lies between 21 and 22. 446 is 5 above 441 and 38 below 484, so the root is closer to 21.
- Straight line between 441 and 484: 21.1163 (0.01% low)
- Tangent from 21, i.e. 21 + 5 ÷ 42: 21.1190 (0% high)
- Tangent from 22, i.e. 22 − 38 ÷ 44: 21.1364 (0.08% high)
For √446 the tangent at 21 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 446 is just 5 above 441.
Finding √446 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 | 446 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 21.2380952381 | 21.1190476190 | 3 |
| 2 | 21.1190476190 | 21.1183765502 | 21.1187120846 | 8 |
| 3 | 21.1187120846 | 21.1187120793 | 21.1187120819 | 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 √446 = 21.1187120819 to every decimal shown.
√446 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √446 the pattern is [21; 8, 2, 2, 1, 3, 1, 1, 20, 1, 1, 3, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √446 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 |
|---|---|---|
| 21/1 | 21.0000000000 | 1.2 × 10⁻¹ |
| 169/8 | 21.1250000000 | 6.3 × 10⁻³ |
| 359/17 | 21.1176470588 | 1.1 × 10⁻³ |
| 887/42 | 21.1190476190 | 3.4 × 10⁻⁴ |
| 1,246/59 | 21.1186440678 | 6.8 × 10⁻⁵ |
| 4,625/219 | 21.1187214612 | 9.4 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 446y² = 1. Its smallest solution in positive whole numbers is x = 110,166,015, y = 5,216,512.
√446 in geometry and everyday measurements
- A square garage floor of 446 square feet measures about 21.12 ft (21 ft 1 in) per side, and its corner-to-corner diagonal is √892 ≈ 29.9 ft.
- 446 is not a sum of two whole-number squares — the prime factor 223 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √446 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 21 box, because 1² + 2² + 21² = 446.
Square roots near √446 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √443 | √443 | 21.0476 | No |
| √444 | 2√111 | 21.0713 | No |
| √445 | √445 | 21.0950 | No |
| √446 | √446 | 21.1187 | No |
| √447 | √447 | 21.1424 | No |
| √448 | 8√7 | 21.1660 | No |
| √449 | √449 | 21.1896 | No |
- The cube root of 446 is about 7.640321.
- Squaring undoes the root: (√446)² = 446, while 446² = 198,916 — the number whose square root is 446.
Frequently asked questions
What is the square root of 446?
The square root of 446 is √446, about 21.1187120819. The negative root, −21.118712, also squares to 446.
Is the square root of 446 rational or irrational?
Irrational. 446 is not a perfect square — it falls between 441 and 484 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √446 be simplified?
No. 446 = 2 × 223 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √446 rounded to two decimal places?
√446 ≈ 21.12 to two decimal places (21.1 to one, 21.119 to three). Check: 21.12² = 446.0544, close to 446.