√114 at a glance
- Exact value
- √114
- Decimal (10 places)
- 10.6770782520
- Rounded
- 10.7 · 10.68 · 10.677
- Perfect square?
- No — between 10² and 11²
- Rational?
- Irrational
- Both square roots
- ±10.677078
- Prime factorization
- 2 × 3 × 19
- Cube root
- 4.848808
How to simplify √114
The prime factorization of 114 is 2 × 3 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √114 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 114, 2, 3 and 19 appear an odd number of times, so √114 is irrational and 10.6770782520 is a rounded value.
Where √114 sits between perfect squares
100 = 10² and 121 = 11² are the nearest perfect squares, so √114 lies between 10 and 11. 114 is 14 above 100 and 7 below 121, so the root is closer to 11.
- Straight line between 100 and 121: 10.6667 (0.1% low)
- Tangent from 10, i.e. 10 + 14 ÷ 20: 10.7000 (0.21% high)
- Tangent from 11, i.e. 11 − 7 ÷ 22: 10.6818 (0.04% high)
For √114 the tangent at 11 wins, missing by only 0.0047. Tangent estimates shine when the number sits close to a perfect square — here 114 is just 7 below 121.
Finding √114 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, 11 (11² = 121):
| Step | Guess x | 114 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 11.0000000000 | 10.3636363636 | 10.6818181818 | 2 |
| 2 | 10.6818181818 | 10.6723404255 | 10.6770793037 | 5 |
| 3 | 10.6770793037 | 10.6770772004 | 10.6770782520 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √114 = 10.6770782520 to every decimal shown.
√114 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √114 the pattern is [10; 1, 2, 10, 2, 1, 20] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √114 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 |
|---|---|---|
| 10/1 | 10.0000000000 | 6.8 × 10⁻¹ |
| 11/1 | 11.0000000000 | 3.2 × 10⁻¹ |
| 32/3 | 10.6666666667 | 1.0 × 10⁻² |
| 331/31 | 10.6774193548 | 3.4 × 10⁻⁴ |
| 694/65 | 10.6769230769 | 1.6 × 10⁻⁴ |
| 1,025/96 | 10.6770833333 | 5.1 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 114y² = 1. Its smallest solution in positive whole numbers is x = 1,025, y = 96.
√114 in geometry and everyday measurements
- A square room or garden bed covering 114 square feet measures about 10.68 ft (10 ft 8 in) along each wall.
- 114 is not a sum of two whole-number squares — the prime factor 3 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √114 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 8 box, because 1² + 7² + 8² = 114.
Square roots near √114 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √111 | √111 | 10.5357 | No |
| √112 | 4√7 | 10.5830 | No |
| √113 | √113 | 10.6301 | No |
| √114 | √114 | 10.6771 | No |
| √115 | √115 | 10.7238 | No |
| √116 | 2√29 | 10.7703 | No |
| √117 | 3√13 | 10.8167 | No |
- The cube root of 114 is about 4.848808.
- Four times the radicand doubles the root: √456 = 2 × √114 ≈ 21.354157.
Frequently asked questions
What is the square root of 114?
The square root of 114 is √114, about 10.6770782520. The negative root, −10.677078, also squares to 114.
Is the square root of 114 rational or irrational?
Irrational. 114 is not a perfect square — it falls between 100 and 121 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √114 be simplified?
No. 114 = 2 × 3 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √114 rounded to two decimal places?
√114 ≈ 10.68 to two decimal places (10.7 to one, 10.677 to three). Check: 10.68² = 114.0624, close to 114.