√34 at a glance
- Exact value
- √34
- Decimal (10 places)
- 5.8309518948
- Rounded
- 5.8 · 5.83 · 5.831
- Perfect square?
- No — between 5² and 6²
- Rational?
- Irrational
- Both square roots
- ±5.830952
- Prime factorization
- 2 × 17
- Cube root
- 3.239612
How to simplify √34
The prime factorization of 34 is 2 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √34 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 34, 2 and 17 appear an odd number of times, so √34 is irrational and 5.8309518948 is a rounded value.
Where √34 sits between perfect squares
25 = 5² and 36 = 6² are the nearest perfect squares, so √34 lies between 5 and 6. 34 is 9 above 25 and 2 below 36, so the root is closer to 6.
- Straight line between 25 and 36: 5.8182 (0.22% low)
- Tangent from 5, i.e. 5 + 9 ÷ 10: 5.9000 (1.18% high)
- Tangent from 6, i.e. 6 − 2 ÷ 12: 5.8333 (0.04% high)
For √34 the tangent at 6 wins, missing by only 0.0024. Tangent estimates shine when the number sits close to a perfect square — here 34 is just 2 below 36.
Finding √34 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, 6 (6² = 36):
| Step | Guess x | 34 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 6.0000000000 | 5.6666666667 | 5.8333333333 | 2 |
| 2 | 5.8333333333 | 5.8285714286 | 5.8309523810 | 6 |
| 3 | 5.8309523810 | 5.8309514087 | 5.8309518948 | 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 √34 = 5.8309518948 to every decimal shown.
√34 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √34 the pattern is [5; 1, 4, 1, 10] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √34 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 |
|---|---|---|
| 5/1 | 5.0000000000 | 8.3 × 10⁻¹ |
| 6/1 | 6.0000000000 | 1.7 × 10⁻¹ |
| 29/5 | 5.8000000000 | 3.1 × 10⁻² |
| 35/6 | 5.8333333333 | 2.4 × 10⁻³ |
| 379/65 | 5.8307692308 | 1.8 × 10⁻⁴ |
| 414/71 | 5.8309859155 | 3.4 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 34y² = 1. Its smallest solution in positive whole numbers is x = 35, y = 6.
√34 in geometry and everyday measurements
- A square room or garden bed covering 34 square feet measures about 5.83 ft (5 ft 10 in) along each wall.
- 34 = 3² + 5², so by the Pythagorean theorem √34 is the diagonal of a 3 × 5 rectangle — and the distance between the points (0, 0) and (3, 5) on a grid.
Square roots near √34 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √31 | √31 | 5.5678 | No |
| √32 | 4√2 | 5.6569 | No |
| √33 | √33 | 5.7446 | No |
| √34 | √34 | 5.8310 | No |
| √35 | √35 | 5.9161 | No |
| √36 | 6 | 6.0000 | Yes |
| √37 | √37 | 6.0828 | No |
- The cube root of 34 is about 3.239612.
- Four times the radicand doubles the root: √136 = 2 × √34 ≈ 11.661904.
Frequently asked questions
What is the square root of 34?
The square root of 34 is √34, about 5.8309518948. The negative root, −5.830952, also squares to 34.
Is the square root of 34 rational or irrational?
Irrational. 34 is not a perfect square — it falls between 25 and 36 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √34 be simplified?
No. 34 = 2 × 17 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √34 rounded to two decimal places?
√34 ≈ 5.83 to two decimal places (5.8 to one, 5.831 to three). Check: 5.83² = 33.9889, close to 34.