√466 at a glance
- Exact value
- √466
- Decimal (10 places)
- 21.5870331449
- Rounded
- 21.6 · 21.59 · 21.587
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.587033
- Prime factorization
- 2 × 233
- Cube root
- 7.752861
How to simplify √466
The prime factorization of 466 is 2 × 233. Every prime appears only once, so there is no pair to bring outside the radical — √466 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 466, 2 and 233 appear an odd number of times, so √466 is irrational and 21.5870331449 is a rounded value.
Where √466 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √466 lies between 21 and 22. 466 is 25 above 441 and 18 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.5814 (0.03% low)
- Tangent from 21, i.e. 21 + 25 ÷ 42: 21.5952 (0.04% high)
- Tangent from 22, i.e. 22 − 18 ÷ 44: 21.5909 (0.02% high)
For √466 the tangent at 22 wins, missing by only 0.0039. Tangent estimates shine when the number sits close to a perfect square — here 466 is just 18 below 484.
Finding √466 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, 22 (22² = 484):
| Step | Guess x | 466 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.1818181818 | 21.5909090909 | 2 |
| 2 | 21.5909090909 | 21.5831578947 | 21.5870334928 | 6 |
| 3 | 21.5870334928 | 21.5870327970 | 21.5870331449 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √466 = 21.5870331449 to every decimal shown.
√466 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √466 the pattern is [21; 1, 1, 2, 2, 1, 2, 5, 1, 3, 1, 20, 1, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √466 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 | 5.9 × 10⁻¹ |
| 22/1 | 22.0000000000 | 4.1 × 10⁻¹ |
| 43/2 | 21.5000000000 | 8.7 × 10⁻² |
| 108/5 | 21.6000000000 | 1.3 × 10⁻² |
| 259/12 | 21.5833333333 | 3.7 × 10⁻³ |
| 367/17 | 21.5882352941 | 1.2 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 466y² = 1. Its smallest solution in positive whole numbers is x = 938,319,425, y = 43,466,808.
√466 in geometry and everyday measurements
- A square garage floor of 466 square feet measures about 21.59 ft (21 ft 7 in) per side, and its corner-to-corner diagonal is √932 ≈ 30.5 ft.
- 466 = 5² + 21², so by the Pythagorean theorem √466 is the diagonal of a 5 × 21 rectangle — and the distance between the points (0, 0) and (5, 21) on a grid.
Square roots near √466 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √463 | √463 | 21.5174 | No |
| √464 | 4√29 | 21.5407 | No |
| √465 | √465 | 21.5639 | No |
| √466 | √466 | 21.5870 | No |
| √467 | √467 | 21.6102 | No |
| √468 | 6√13 | 21.6333 | No |
| √469 | √469 | 21.6564 | No |
- The cube root of 466 is about 7.752861.
- Squaring undoes the root: (√466)² = 466, while 466² = 217,156 — the number whose square root is 466.
Frequently asked questions
What is the square root of 466?
The square root of 466 is √466, about 21.5870331449. The negative root, −21.587033, also squares to 466.
Is the square root of 466 rational or irrational?
Irrational. 466 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 √466 be simplified?
No. 466 = 2 × 233 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √466 rounded to two decimal places?
√466 ≈ 21.59 to two decimal places (21.6 to one, 21.587 to three). Check: 21.59² = 466.1281, close to 466.