√134 at a glance
- Exact value
- √134
- Decimal (10 places)
- 11.5758369028
- Rounded
- 11.6 · 11.58 · 11.576
- Perfect square?
- No — between 11² and 12²
- Rational?
- Irrational
- Both square roots
- ±11.575837
- Prime factorization
- 2 × 67
- Cube root
- 5.117230
How to simplify √134
The prime factorization of 134 is 2 × 67. Every prime appears only once, so there is no pair to bring outside the radical — √134 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 134, 2 and 67 appear an odd number of times, so √134 is irrational and 11.5758369028 is a rounded value.
Where √134 sits between perfect squares
121 = 11² and 144 = 12² are the nearest perfect squares, so √134 lies between 11 and 12. 134 is 13 above 121 and 10 below 144, so the root is closer to 12.
- Straight line between 121 and 144: 11.5652 (0.09% low)
- Tangent from 11, i.e. 11 + 13 ÷ 22: 11.5909 (0.13% high)
- Tangent from 12, i.e. 12 − 10 ÷ 24: 11.5833 (0.06% high)
For √134 the tangent at 12 wins, missing by only 0.0075. Tangent estimates shine when the number sits close to a perfect square — here 134 is just 10 below 144.
Finding √134 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, 12 (12² = 144):
| Step | Guess x | 134 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 12.0000000000 | 11.1666666667 | 11.5833333333 | 2 |
| 2 | 11.5833333333 | 11.5683453237 | 11.5758393285 | 5 |
| 3 | 11.5758393285 | 11.5758344770 | 11.5758369028 | 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 √134 = 11.5758369028 to every decimal shown.
√134 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √134 the pattern is [11; 1, 1, 2, 1, 3, 1, 10, 1, 3, 1, 2, 1, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √134 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 |
|---|---|---|
| 11/1 | 11.0000000000 | 5.8 × 10⁻¹ |
| 12/1 | 12.0000000000 | 4.2 × 10⁻¹ |
| 23/2 | 11.5000000000 | 7.6 × 10⁻² |
| 58/5 | 11.6000000000 | 2.4 × 10⁻² |
| 81/7 | 11.5714285714 | 4.4 × 10⁻³ |
| 301/26 | 11.5769230769 | 1.1 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 134y² = 1. Its smallest solution in positive whole numbers is x = 145,925, y = 12,606.
√134 in geometry and everyday measurements
- A square room or garden bed covering 134 square feet measures about 11.58 ft (11 ft 7 in) along each wall.
- 134 is not a sum of two whole-number squares — the prime factor 67 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √134 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 3 × 11 box, because 2² + 3² + 11² = 134.
Square roots near √134 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √131 | √131 | 11.4455 | No |
| √132 | 2√33 | 11.4891 | No |
| √133 | √133 | 11.5326 | No |
| √134 | √134 | 11.5758 | No |
| √135 | 3√15 | 11.6190 | No |
| √136 | 2√34 | 11.6619 | No |
| √137 | √137 | 11.7047 | No |
- The cube root of 134 is about 5.117230.
- Four times the radicand doubles the root: √536 = 2 × √134 ≈ 23.151674.
Frequently asked questions
What is the square root of 134?
The square root of 134 is √134, about 11.5758369028. The negative root, −11.575837, also squares to 134.
Is the square root of 134 rational or irrational?
Irrational. 134 is not a perfect square — it falls between 121 and 144 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √134 be simplified?
No. 134 = 2 × 67 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √134 rounded to two decimal places?
√134 ≈ 11.58 to two decimal places (11.6 to one, 11.576 to three). Check: 11.58² = 134.0964, close to 134.