√444 at a glance
- Exact value
- 2√111
- Decimal (10 places)
- 21.0713075057
- Rounded
- 21.1 · 21.07 · 21.071
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.071308
- Prime factorization
- 2² × 3 × 37
- Cube root
- 7.628884
How to simplify √444
Look for the largest perfect square that divides 444. Here it is 4 (2²), because 444 = 4 × 111 and 111 has no square factor left:
The prime factorization tells the same story: 444 = 2² × 3 × 37. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 37 stays inside.
Check: (2√111)² = 2² × 111 = 4 × 111 = 444. As a decimal, 2√111 = 2 × 10.5356537529 ≈ 21.0713075057.
Where √444 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √444 lies between 21 and 22. 444 is 3 above 441 and 40 below 484, so the root is closer to 21.
- Straight line between 441 and 484: 21.0698 (0.01% low)
- Tangent from 21, i.e. 21 + 3 ÷ 42: 21.0714 (0% high)
- Tangent from 22, i.e. 22 − 40 ÷ 44: 21.0909 (0.09% high)
For √444 the tangent at 21 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 444 is just 3 above 441.
Finding √444 with the Babylonian method
Picture a rectangle with an area of 444 and one side x; the other side must be 444 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √444.
Start from the nearest whole number, 21 (21² = 441):
| Step | Guess x | 444 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 21.1428571429 | 21.0714285714 | 3 |
| 2 | 21.0714285714 | 21.0711864407 | 21.0713075061 | 9 |
| 3 | 21.0713075061 | 21.0713075054 | 21.0713075057 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √444 = 21.0713075057 to every decimal shown.
√444 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √444 the pattern is [21; 14, 42] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √444 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 | 7.1 × 10⁻² |
| 295/14 | 21.0714285714 | 1.2 × 10⁻⁴ |
| 12,411/589 | 21.0713073005 | 2.1 × 10⁻⁷ |
| 174,049/8,260 | 21.0713075061 | 3.5 × 10⁻¹⁰ |
| 7,322,469/347,509 | 21.0713075057 | < 10⁻¹⁰ |
| 102,688,615/4,873,386 | 21.0713075057 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 444y² = 1. Its smallest solution in positive whole numbers is x = 295, y = 14.
√444 in geometry and everyday measurements
- A square garage floor of 444 square feet measures about 21.07 ft (21 ft 1 in) per side, and its corner-to-corner diagonal is √888 ≈ 29.8 ft.
- 444 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √444 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √444 as its space diagonal.
- Since √444 = 2√111, a length of √444 is exactly 2 copies of the length √111 laid end to end.
Square roots near √444 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √441 | 21 | 21.0000 | Yes |
| √442 | √442 | 21.0238 | No |
| √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 |
- The cube root of 444 is about 7.628884.
- Because 444 = 4 × 111, the root is twice √111: 2 × 10.535654 ≈ 21.071308.
Frequently asked questions
What is the square root of 444?
The square root of 444 is 2√111 in simplest radical form, which is about 21.0713075057. The negative root, −21.071308, also squares to 444.
Is the square root of 444 rational or irrational?
Irrational. 444 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 √444 be simplified?
Yes. The largest perfect square dividing 444 is 4, so √444 = √4 × √111 = 2√111.
What is √444 rounded to two decimal places?
√444 ≈ 21.07 to two decimal places (21.1 to one, 21.071 to three). Check: 21.07² = 443.9449, close to 444.