√449 at a glance
- Exact value
- √449
- Decimal (10 places)
- 21.1896201004
- Rounded
- 21.2 · 21.19 · 21.190
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.189620
- Prime factorization
- 449
- Cube root
- 7.657414
How to simplify √449
449 is a prime number, so its only factors are 1 and 449. There is no perfect-square factor to pull out, which means √449 is already in its simplest radical form.
The square root of any prime is irrational. If √449 were a fraction a/b in lowest terms, then a² = 449b², so 449 would divide a — and then 449 would divide b too, contradicting “lowest terms.” That is why the decimal 21.1896201004 is only a rounded value.
Where √449 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √449 lies between 21 and 22. 449 is 8 above 441 and 35 below 484, so the root is closer to 21.
- Straight line between 441 and 484: 21.1860 (0.02% low)
- Tangent from 21, i.e. 21 + 8 ÷ 42: 21.1905 (0% high)
- Tangent from 22, i.e. 22 − 35 ÷ 44: 21.2045 (0.07% high)
For √449 the tangent at 21 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 449 is just 8 above 441.
Finding √449 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 449: following the tangent line down to zero simplifies to averaging x with 449 ÷ x.
Start from the nearest whole number, 21 (21² = 441):
| Step | Guess x | 449 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 21.3809523810 | 21.1904761905 | 3 |
| 2 | 21.1904761905 | 21.1887640449 | 21.1896201177 | 7 |
| 3 | 21.1896201177 | 21.1896200831 | 21.1896201004 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √449 = 21.1896201004 to every decimal shown.
√449 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √449 the pattern is [21; 5, 3, 1, 1, 1, 7, 1, 5, 5, 1, 7, 1, …] with the block of 17 terms after the semicolon repeating forever (only the first 12 of the 17 are shown). A pattern that never ends is one more proof that √449 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 | 1.9 × 10⁻¹ |
| 106/5 | 21.2000000000 | 1.0 × 10⁻² |
| 339/16 | 21.1875000000 | 2.1 × 10⁻³ |
| 445/21 | 21.1904761905 | 8.6 × 10⁻⁴ |
| 784/37 | 21.1891891892 | 4.3 × 10⁻⁴ |
| 1,229/58 | 21.1896551724 | 3.5 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 449y² = 1. Its smallest solution in positive whole numbers is x = 71,798,771,299,708,449, y = 3,388,393,513,402,120 — 17 digits for x, even though 449 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 189,471,332² − 449 × 8,941,705² = −1.
√449 in geometry and everyday measurements
- A square garage floor of 449 square feet measures about 21.19 ft (21 ft 2 in) per side, and its corner-to-corner diagonal is √898 ≈ 30 ft.
- 449 = 7² + 20², so by the Pythagorean theorem √449 is the diagonal of a 7 × 20 rectangle — and the distance between the points (0, 0) and (7, 20) on a grid.
Square roots near √449 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √446 | √446 | 21.1187 | No |
| √447 | √447 | 21.1424 | No |
| √448 | 8√7 | 21.1660 | No |
| √449 | √449 | 21.1896 | No |
| √450 | 15√2 | 21.2132 | No |
| √451 | √451 | 21.2368 | No |
| √452 | 2√113 | 21.2603 | No |
- The cube root of 449 is about 7.657414.
- Squaring undoes the root: (√449)² = 449, while 449² = 201,601 — the number whose square root is 449.
Frequently asked questions
What is the square root of 449?
The square root of 449 is √449, about 21.1896201004. The negative root, −21.189620, also squares to 449.
Is the square root of 449 rational or irrational?
Irrational. 449 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 √449 be simplified?
No. 449 is prime, so there is no perfect square to take out of the radical.
What is √449 rounded to two decimal places?
√449 ≈ 21.19 to two decimal places (21.2 to one, 21.190 to three). Check: 21.19² = 449.0161, close to 449.