√186 at a glance
- Exact value
- √186
- Decimal (10 places)
- 13.6381816970
- Rounded
- 13.6 · 13.64 · 13.638
- Perfect square?
- No — between 13² and 14²
- Rational?
- Irrational
- Both square roots
- ±13.638182
- Prime factorization
- 2 × 3 × 31
- Cube root
- 5.708267
How to simplify √186
The prime factorization of 186 is 2 × 3 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √186 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 186, 2, 3 and 31 appear an odd number of times, so √186 is irrational and 13.6381816970 is a rounded value.
Where √186 sits between perfect squares
169 = 13² and 196 = 14² are the nearest perfect squares, so √186 lies between 13 and 14. 186 is 17 above 169 and 10 below 196, so the root is closer to 14.
- Straight line between 169 and 196: 13.6296 (0.06% low)
- Tangent from 13, i.e. 13 + 17 ÷ 26: 13.6538 (0.11% high)
- Tangent from 14, i.e. 14 − 10 ÷ 28: 13.6429 (0.03% high)
For √186 the tangent at 14 wins, missing by only 0.0047. Tangent estimates shine when the number sits close to a perfect square — here 186 is just 10 below 196.
Finding √186 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, 14 (14² = 196):
| Step | Guess x | 186 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 14.0000000000 | 13.2857142857 | 13.6428571429 | 2 |
| 2 | 13.6428571429 | 13.6335078534 | 13.6381824981 | 6 |
| 3 | 13.6381824981 | 13.6381808958 | 13.6381816970 | 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 √186 = 13.6381816970 to every decimal shown.
√186 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √186 the pattern is [13; 1, 1, 1, 3, 4, 3, 1, 1, 1, 26] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √186 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 |
|---|---|---|
| 13/1 | 13.0000000000 | 6.4 × 10⁻¹ |
| 14/1 | 14.0000000000 | 3.6 × 10⁻¹ |
| 27/2 | 13.5000000000 | 1.4 × 10⁻¹ |
| 41/3 | 13.6666666667 | 2.8 × 10⁻² |
| 150/11 | 13.6363636364 | 1.8 × 10⁻³ |
| 641/47 | 13.6382978723 | 1.2 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 186y² = 1. Its smallest solution in positive whole numbers is x = 7,501, y = 550.
√186 in geometry and everyday measurements
- A square room or garden bed covering 186 square feet measures about 13.64 ft (13 ft 8 in) along each wall.
- 186 is not a sum of two whole-number squares — the prime factor 3 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √186 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 13 box, because 1² + 4² + 13² = 186.
Square roots near √186 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √183 | √183 | 13.5277 | No |
| √184 | 2√46 | 13.5647 | No |
| √185 | √185 | 13.6015 | No |
| √186 | √186 | 13.6382 | No |
| √187 | √187 | 13.6748 | No |
| √188 | 2√47 | 13.7113 | No |
| √189 | 3√21 | 13.7477 | No |
- The cube root of 186 is about 5.708267.
- Four times the radicand doubles the root: √744 = 2 × √186 ≈ 27.276363.
Frequently asked questions
What is the square root of 186?
The square root of 186 is √186, about 13.6381816970. The negative root, −13.638182, also squares to 186.
Is the square root of 186 rational or irrational?
Irrational. 186 is not a perfect square — it falls between 169 and 196 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √186 be simplified?
No. 186 = 2 × 3 × 31 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √186 rounded to two decimal places?
√186 ≈ 13.64 to two decimal places (13.6 to one, 13.638 to three). Check: 13.64² = 186.0496, close to 186.