√66 at a glance
- Exact value
- √66
- Decimal (10 places)
- 8.1240384046
- Rounded
- 8.1 · 8.12 · 8.124
- Perfect square?
- No — between 8² and 9²
- Rational?
- Irrational
- Both square roots
- ±8.124038
- Prime factorization
- 2 × 3 × 11
- Cube root
- 4.041240
How to simplify √66
The prime factorization of 66 is 2 × 3 × 11. Every prime appears only once, so there is no pair to bring outside the radical — √66 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 66, 2, 3 and 11 appear an odd number of times, so √66 is irrational and 8.1240384046 is a rounded value.
Where √66 sits between perfect squares
64 = 8² and 81 = 9² are the nearest perfect squares, so √66 lies between 8 and 9. 66 is 2 above 64 and 15 below 81, so the root is closer to 8.
- Straight line between 64 and 81: 8.1176 (0.08% low)
- Tangent from 8, i.e. 8 + 2 ÷ 16: 8.1250 (0.01% high)
- Tangent from 9, i.e. 9 − 15 ÷ 18: 8.1667 (0.52% high)
For √66 the tangent at 8 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 66 is just 2 above 64.
Finding √66 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, 8 (8² = 64):
| Step | Guess x | 66 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 8.0000000000 | 8.2500000000 | 8.1250000000 | 3 |
| 2 | 8.1250000000 | 8.1230769231 | 8.1240384615 | 7 |
| 3 | 8.1240384615 | 8.1240383477 | 8.1240384046 | 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 √66 = 8.1240384046 to every decimal shown.
√66 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √66 the pattern is [8; 8, 16] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √66 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 |
|---|---|---|
| 8/1 | 8.0000000000 | 1.2 × 10⁻¹ |
| 65/8 | 8.1250000000 | 9.6 × 10⁻⁴ |
| 1,048/129 | 8.1240310078 | 7.4 × 10⁻⁶ |
| 8,449/1,040 | 8.1240384615 | 5.7 × 10⁻⁸ |
| 136,232/16,769 | 8.1240384042 | 4.4 × 10⁻¹⁰ |
| 1,098,305/135,192 | 8.1240384046 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 66y² = 1. Its smallest solution in positive whole numbers is x = 65, y = 8.
√66 in geometry and everyday measurements
- A square room or garden bed covering 66 square feet measures about 8.12 ft (8 ft 1 in) along each wall.
- 66 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √66 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 8 box, because 1² + 1² + 8² = 66.
Square roots near √66 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √63 | 3√7 | 7.9373 | No |
| √64 | 8 | 8.0000 | Yes |
| √65 | √65 | 8.0623 | No |
| √66 | √66 | 8.1240 | No |
| √67 | √67 | 8.1854 | No |
| √68 | 2√17 | 8.2462 | No |
| √69 | √69 | 8.3066 | No |
- The cube root of 66 is about 4.041240.
- Four times the radicand doubles the root: √264 = 2 × √66 ≈ 16.248077.
Frequently asked questions
What is the square root of 66?
The square root of 66 is √66, about 8.1240384046. The negative root, −8.124038, also squares to 66.
Is the square root of 66 rational or irrational?
Irrational. 66 is not a perfect square — it falls between 64 and 81 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √66 be simplified?
No. 66 = 2 × 3 × 11 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √66 rounded to two decimal places?
√66 ≈ 8.12 to two decimal places (8.1 to one, 8.124 to three). Check: 8.12² = 65.9344, close to 66.