√666 at a glance
- Exact value
- 3√74
- Decimal (10 places)
- 25.8069758011
- Rounded
- 25.8 · 25.81 · 25.807
- Perfect square?
- No — between 25² and 26²
- Rational?
- Irrational
- Both square roots
- ±25.806976
- Prime factorization
- 2 × 3² × 37
- Cube root
- 8.732892
How to simplify √666
Look for the largest perfect square that divides 666. Here it is 9 (3²), because 666 = 9 × 74 and 74 has no square factor left:
The prime factorization tells the same story: 666 = 2 × 3² × 37. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 37 stays inside.
Check: (3√74)² = 3² × 74 = 9 × 74 = 666. As a decimal, 3√74 = 3 × 8.602325267 ≈ 25.8069758011.
Where √666 sits between perfect squares
625 = 25² and 676 = 26² are the nearest perfect squares, so √666 lies between 25 and 26. 666 is 41 above 625 and 10 below 676, so the root is closer to 26.
- Straight line between 625 and 676: 25.8039 (0.01% low)
- Tangent from 25, i.e. 25 + 41 ÷ 50: 25.8200 (0.05% high)
- Tangent from 26, i.e. 26 − 10 ÷ 52: 25.8077 (0% high)
For √666 the tangent at 26 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 666 is just 10 below 676.
Finding √666 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, 26 (26² = 676):
| Step | Guess x | 666 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 26.0000000000 | 25.6153846154 | 25.8076923077 | 3 |
| 2 | 25.8076923077 | 25.8062593145 | 25.8069758111 | 8 |
| 3 | 25.8069758111 | 25.8069757912 | 25.8069758011 | 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 √666 = 25.8069758011 to every decimal shown.
√666 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √666 the pattern is [25; 1, 4, 5, 1, 1, 6, 1, 4, 1, 6, 1, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √666 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 |
|---|---|---|
| 25/1 | 25.0000000000 | 8.1 × 10⁻¹ |
| 26/1 | 26.0000000000 | 1.9 × 10⁻¹ |
| 129/5 | 25.8000000000 | 7.0 × 10⁻³ |
| 671/26 | 25.8076923077 | 7.2 × 10⁻⁴ |
| 800/31 | 25.8064516129 | 5.2 × 10⁻⁴ |
| 1,471/57 | 25.8070175439 | 4.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 666y² = 1. Its smallest solution in positive whole numbers is x = 27,365,201, y = 1,060,380.
√666 in geometry and everyday measurements
- 666 square feet is 61.9 m². Laid out as a square — a small house footprint or a lot — it is about 25.81 ft (25 ft 10 in) on a side.
- 666 = 15² + 21², so by the Pythagorean theorem √666 is the diagonal of a 15 × 21 rectangle — and the distance between the points (0, 0) and (15, 21) on a grid.
- Since √666 = 3√74, a length of √666 is exactly 3 copies of the length √74 laid end to end.
Square roots near √666 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √663 | √663 | 25.7488 | No |
| √664 | 2√166 | 25.7682 | No |
| √665 | √665 | 25.7876 | No |
| √666 | 3√74 | 25.8070 | No |
| √667 | √667 | 25.8263 | No |
| √668 | 2√167 | 25.8457 | No |
| √669 | √669 | 25.8650 | No |
- The cube root of 666 is about 8.732892.
- Squaring undoes the root: (√666)² = 666, while 666² = 443,556 — the number whose square root is 666.
Frequently asked questions
What is the square root of 666?
The square root of 666 is 3√74 in simplest radical form, which is about 25.8069758011. The negative root, −25.806976, also squares to 666.
Is the square root of 666 rational or irrational?
Irrational. 666 is not a perfect square — it falls between 625 and 676 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √666 be simplified?
Yes. The largest perfect square dividing 666 is 9, so √666 = √9 × √74 = 3√74.
What is √666 rounded to two decimal places?
√666 ≈ 25.81 to two decimal places (25.8 to one, 25.807 to three). Check: 25.81² = 666.1561, close to 666.