√447 at a glance
- Exact value
- √447
- Decimal (10 places)
- 21.1423745119
- Rounded
- 21.1 · 21.14 · 21.142
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.142375
- Prime factorization
- 3 × 149
- Cube root
- 7.646027
How to simplify √447
The prime factorization of 447 is 3 × 149. Every prime appears only once, so there is no pair to bring outside the radical — √447 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 447, 3 and 149 appear an odd number of times, so √447 is irrational and 21.1423745119 is a rounded value.
Where √447 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √447 lies between 21 and 22. 447 is 6 above 441 and 37 below 484, so the root is closer to 21.
- Straight line between 441 and 484: 21.1395 (0.01% low)
- Tangent from 21, i.e. 21 + 6 ÷ 42: 21.1429 (0% high)
- Tangent from 22, i.e. 22 − 37 ÷ 44: 21.1591 (0.08% high)
For √447 the tangent at 21 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 447 is just 6 above 441.
Finding √447 with the Babylonian method
If a guess is too big, 447 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√447) in one step.
Start from the nearest whole number, 21 (21² = 441):
| Step | Guess x | 447 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 21.2857142857 | 21.1428571429 | 3 |
| 2 | 21.1428571429 | 21.1418918919 | 21.1423745174 | 8 |
| 3 | 21.1423745174 | 21.1423745064 | 21.1423745119 | 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 √447 = 21.1423745119 to every decimal shown.
√447 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √447 the pattern is [21; 7, 42] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √447 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 | 1.4 × 10⁻¹ |
| 148/7 | 21.1428571429 | 4.8 × 10⁻⁴ |
| 6,237/295 | 21.1423728814 | 1.6 × 10⁻⁶ |
| 43,807/2,072 | 21.1423745174 | 5.5 × 10⁻⁹ |
| 1,846,131/87,319 | 21.1423745118 | < 10⁻¹⁰ |
| 12,966,724/613,305 | 21.1423745119 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 447y² = 1. Its smallest solution in positive whole numbers is x = 148, y = 7.
√447 in geometry and everyday measurements
- A square garage floor of 447 square feet measures about 21.14 ft (21 ft 2 in) per side, and its corner-to-corner diagonal is √894 ≈ 29.9 ft.
- 447 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 √447 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 √447 as its space diagonal.
Square roots near √447 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √450 | 15√2 | 21.2132 | No |
- The cube root of 447 is about 7.646027.
- Squaring undoes the root: (√447)² = 447, while 447² = 199,809 — the number whose square root is 447.
Frequently asked questions
What is the square root of 447?
The square root of 447 is √447, about 21.1423745119. The negative root, −21.142375, also squares to 447.
Is the square root of 447 rational or irrational?
Irrational. 447 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 √447 be simplified?
No. 447 = 3 × 149 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √447 rounded to two decimal places?
√447 ≈ 21.14 to two decimal places (21.1 to one, 21.142 to three). Check: 21.14² = 446.8996, close to 447.