√445 at a glance
- Exact value
- √445
- Decimal (10 places)
- 21.0950231097
- Rounded
- 21.1 · 21.10 · 21.095
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.095023
- Prime factorization
- 5 × 89
- Cube root
- 7.634607
How to simplify √445
The prime factorization of 445 is 5 × 89. Every prime appears only once, so there is no pair to bring outside the radical — √445 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 445, 5 and 89 appear an odd number of times, so √445 is irrational and 21.0950231097 is a rounded value.
Where √445 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √445 lies between 21 and 22. 445 is 4 above 441 and 39 below 484, so the root is closer to 21.
- Straight line between 441 and 484: 21.0930 (0.01% low)
- Tangent from 21, i.e. 21 + 4 ÷ 42: 21.0952 (0% high)
- Tangent from 22, i.e. 22 − 39 ÷ 44: 21.1136 (0.09% high)
For √445 the tangent at 21 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 445 is just 4 above 441.
Finding √445 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 445: following the tangent line down to zero simplifies to averaging x with 445 ÷ x.
Start from the nearest whole number, 21 (21² = 441):
| Step | Guess x | 445 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 21.1904761905 | 21.0952380952 | 3 |
| 2 | 21.0952380952 | 21.0948081264 | 21.0950231108 | 8 |
| 3 | 21.0950231108 | 21.0950231086 | 21.0950231097 | 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 √445 = 21.0950231097 to every decimal shown.
√445 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √445 the pattern is [21; 10, 1, 1, 10, 42] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √445 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 | 9.5 × 10⁻² |
| 211/10 | 21.1000000000 | 5.0 × 10⁻³ |
| 232/11 | 21.0909090909 | 4.1 × 10⁻³ |
| 443/21 | 21.0952380952 | 2.1 × 10⁻⁴ |
| 4,662/221 | 21.0950226244 | 4.9 × 10⁻⁷ |
| 196,247/9,303 | 21.0950231108 | 1.1 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 445y² = 1. Its smallest solution in positive whole numbers is x = 43,468,489, y = 2,060,604. Because the period is odd, the equation with −1 on the right also has a solution: 4,662² − 445 × 221² = −1.
√445 in geometry and everyday measurements
- A square garage floor of 445 square feet measures about 21.1 ft (21 ft 1 in) per side, and its corner-to-corner diagonal is √890 ≈ 29.8 ft.
- 445 = 2² + 21² = 11² + 18², so by the Pythagorean theorem √445 is the diagonal of rectangles measuring 2 × 21 and 11 × 18 — and the distance between the points (0, 0) and (2, 21) on a grid.
Square roots near √445 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √448 | 8√7 | 21.1660 | No |
- The cube root of 445 is about 7.634607.
- Squaring undoes the root: (√445)² = 445, while 445² = 198,025 — the number whose square root is 445.
Frequently asked questions
What is the square root of 445?
The square root of 445 is √445, about 21.0950231097. The negative root, −21.095023, also squares to 445.
Is the square root of 445 rational or irrational?
Irrational. 445 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 √445 be simplified?
No. 445 = 5 × 89 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √445 rounded to two decimal places?
√445 ≈ 21.10 to two decimal places (21.1 to one, 21.095 to three). Check: 21.10² = 445.21, close to 445.