√166 at a glance
- Exact value
- √166
- Decimal (10 places)
- 12.8840987267
- Rounded
- 12.9 · 12.88 · 12.884
- Perfect square?
- No — between 12² and 13²
- Rational?
- Irrational
- Both square roots
- ±12.884099
- Prime factorization
- 2 × 83
- Cube root
- 5.495865
How to simplify √166
The prime factorization of 166 is 2 × 83. Every prime appears only once, so there is no pair to bring outside the radical — √166 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 166, 2 and 83 appear an odd number of times, so √166 is irrational and 12.8840987267 is a rounded value.
Where √166 sits between perfect squares
144 = 12² and 169 = 13² are the nearest perfect squares, so √166 lies between 12 and 13. 166 is 22 above 144 and 3 below 169, so the root is closer to 13.
- Straight line between 144 and 169: 12.8800 (0.03% low)
- Tangent from 12, i.e. 12 + 22 ÷ 24: 12.9167 (0.25% high)
- Tangent from 13, i.e. 13 − 3 ÷ 26: 12.8846 (0% high)
For √166 the tangent at 13 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 166 is just 3 below 169.
Finding √166 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, 13 (13² = 169):
| Step | Guess x | 166 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 13.0000000000 | 12.7692307692 | 12.8846153846 | 3 |
| 2 | 12.8846153846 | 12.8835820896 | 12.8840987371 | 7 |
| 3 | 12.8840987371 | 12.8840987164 | 12.8840987267 | 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 √166 = 12.8840987267 to every decimal shown.
√166 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √166 the pattern is [12; 1, 7, 1, 1, 1, 2, 4, 1, 3, 2, 12, 2, …] 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 √166 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 |
|---|---|---|
| 12/1 | 12.0000000000 | 8.8 × 10⁻¹ |
| 13/1 | 13.0000000000 | 1.2 × 10⁻¹ |
| 103/8 | 12.8750000000 | 9.1 × 10⁻³ |
| 116/9 | 12.8888888889 | 4.8 × 10⁻³ |
| 219/17 | 12.8823529412 | 1.7 × 10⁻³ |
| 335/26 | 12.8846153846 | 5.2 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 166y² = 1. Its smallest solution in positive whole numbers is x = 1,700,902,565, y = 132,015,642.
√166 in geometry and everyday measurements
- A square room or garden bed covering 166 square feet measures about 12.88 ft (12 ft 11 in) along each wall.
- 166 is not a sum of two whole-number squares — the prime factor 83 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √166 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 9 × 9 box, because 2² + 9² + 9² = 166.
Square roots near √166 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √163 | √163 | 12.7671 | No |
| √164 | 2√41 | 12.8062 | No |
| √165 | √165 | 12.8452 | No |
| √166 | √166 | 12.8841 | No |
| √167 | √167 | 12.9228 | No |
| √168 | 2√42 | 12.9615 | No |
| √169 | 13 | 13.0000 | Yes |
- The cube root of 166 is about 5.495865.
- Four times the radicand doubles the root: √664 = 2 × √166 ≈ 25.768197.
Frequently asked questions
What is the square root of 166?
The square root of 166 is √166, about 12.8840987267. The negative root, −12.884099, also squares to 166.
Is the square root of 166 rational or irrational?
Irrational. 166 is not a perfect square — it falls between 144 and 169 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √166 be simplified?
No. 166 = 2 × 83 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √166 rounded to two decimal places?
√166 ≈ 12.88 to two decimal places (12.9 to one, 12.884 to three). Check: 12.88² = 165.8944, close to 166.