√187 at a glance
- Exact value
- √187
- Decimal (10 places)
- 13.6747943312
- Rounded
- 13.7 · 13.67 · 13.675
- Perfect square?
- No — between 13² and 14²
- Rational?
- Irrational
- Both square roots
- ±13.674794
- Prime factorization
- 11 × 17
- Cube root
- 5.718479
How to simplify √187
The prime factorization of 187 is 11 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √187 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 187, 11 and 17 appear an odd number of times, so √187 is irrational and 13.6747943312 is a rounded value.
Where √187 sits between perfect squares
169 = 13² and 196 = 14² are the nearest perfect squares, so √187 lies between 13 and 14. 187 is 18 above 169 and 9 below 196, so the root is closer to 14.
- Straight line between 169 and 196: 13.6667 (0.06% low)
- Tangent from 13, i.e. 13 + 18 ÷ 26: 13.6923 (0.13% high)
- Tangent from 14, i.e. 14 − 9 ÷ 28: 13.6786 (0.03% high)
For √187 the tangent at 14 wins, missing by only 0.0038. Tangent estimates shine when the number sits close to a perfect square — here 187 is just 9 below 196.
Finding √187 with the Babylonian method
If a guess is too big, 187 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√187) in one step.
Start from the nearest whole number, 14 (14² = 196):
| Step | Guess x | 187 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 14.0000000000 | 13.3571428571 | 13.6785714286 | 2 |
| 2 | 13.6785714286 | 13.6710182768 | 13.6747948527 | 6 |
| 3 | 13.6747948527 | 13.6747938097 | 13.6747943312 | 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 √187 = 13.6747943312 to every decimal shown.
√187 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √187 the pattern is [13; 1, 2, 13, 2, 1, 26] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √187 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 | 6.7 × 10⁻¹ |
| 14/1 | 14.0000000000 | 3.3 × 10⁻¹ |
| 41/3 | 13.6666666667 | 8.1 × 10⁻³ |
| 547/40 | 13.6750000000 | 2.1 × 10⁻⁴ |
| 1,135/83 | 13.6746987952 | 9.6 × 10⁻⁵ |
| 1,682/123 | 13.6747967480 | 2.4 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 187y² = 1. Its smallest solution in positive whole numbers is x = 1,682, y = 123.
√187 in geometry and everyday measurements
- A square room or garden bed covering 187 square feet measures about 13.67 ft (13 ft 8 in) along each wall.
- 187 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √187 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 3 × 13 box, because 3² + 3² + 13² = 187.
Square roots near √187 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √190 | √190 | 13.7840 | No |
- The cube root of 187 is about 5.718479.
- Four times the radicand doubles the root: √748 = 2 × √187 ≈ 27.349589.
Frequently asked questions
What is the square root of 187?
The square root of 187 is √187, about 13.6747943312. The negative root, −13.674794, also squares to 187.
Is the square root of 187 rational or irrational?
Irrational. 187 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 √187 be simplified?
No. 187 = 11 × 17 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √187 rounded to two decimal places?
√187 ≈ 13.67 to two decimal places (13.7 to one, 13.675 to three). Check: 13.67² = 186.8689, close to 187.