√850 at a glance
- Exact value
- 5√34
- Decimal (10 places)
- 29.1547594742
- Rounded
- 29.2 · 29.15 · 29.155
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.154759
- Prime factorization
- 2 × 5² × 17
- Cube root
- 9.472682
How to simplify √850
Look for the largest perfect square that divides 850. Here it is 25 (5²), because 850 = 25 × 34 and 34 has no square factor left:
The prime factorization tells the same story: 850 = 2 × 5² × 17. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 2 × 17 stays inside.
Check: (5√34)² = 5² × 34 = 25 × 34 = 850. As a decimal, 5√34 = 5 × 5.8309518949 ≈ 29.1547594742.
Where √850 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √850 lies between 29 and 30. 850 is 9 above 841 and 50 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.1525 (0.01% low)
- Tangent from 29, i.e. 29 + 9 ÷ 58: 29.1552 (0% high)
- Tangent from 30, i.e. 30 − 50 ÷ 60: 29.1667 (0.04% high)
For √850 the tangent at 29 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 850 is just 9 above 841.
Finding √850 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, 29 (29² = 841):
| Step | Guess x | 850 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.3103448276 | 29.1551724138 | 3 |
| 2 | 29.1551724138 | 29.1543465405 | 29.1547594772 | 8 |
| 3 | 29.1547594772 | 29.1547594713 | 29.1547594742 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √850 = 29.1547594742 to every decimal shown.
√850 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √850 the pattern is [29; 6, 2, 6, 58] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √850 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 |
|---|---|---|
| 29/1 | 29.0000000000 | 1.5 × 10⁻¹ |
| 175/6 | 29.1666666667 | 1.2 × 10⁻² |
| 379/13 | 29.1538461538 | 9.1 × 10⁻⁴ |
| 2,449/84 | 29.1547619048 | 2.4 × 10⁻⁶ |
| 142,421/4,885 | 29.1547594678 | 6.5 × 10⁻⁹ |
| 856,975/29,394 | 29.1547594747 | 5.0 × 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 850y² = 1. Its smallest solution in positive whole numbers is x = 2,449, y = 84.
√850 in geometry and everyday measurements
- 850 square feet is 79 m². Laid out as a square — a small house footprint or a lot — it is about 29.15 ft (29 ft 2 in) on a side.
- 850 = 3² + 29² = 11² + 27² = 15² + 25², so by the Pythagorean theorem √850 is the diagonal of rectangles measuring 3 × 29, 11 × 27 and 15 × 25 — and the distance between the points (0, 0) and (3, 29) on a grid.
- Since √850 = 5√34, a length of √850 is exactly 5 copies of the length √34 laid end to end.
Square roots near √850 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √847 | 11√7 | 29.1033 | No |
| √848 | 4√53 | 29.1204 | No |
| √849 | √849 | 29.1376 | No |
| √850 | 5√34 | 29.1548 | No |
| √851 | √851 | 29.1719 | No |
| √852 | 2√213 | 29.1890 | No |
| √853 | √853 | 29.2062 | No |
- The cube root of 850 is about 9.472682.
- Squaring undoes the root: (√850)² = 850, while 850² = 722,500 — the number whose square root is 850.
Frequently asked questions
What is the square root of 850?
The square root of 850 is 5√34 in simplest radical form, which is about 29.1547594742. The negative root, −29.154759, also squares to 850.
Is the square root of 850 rational or irrational?
Irrational. 850 is not a perfect square — it falls between 841 and 900 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √850 be simplified?
Yes. The largest perfect square dividing 850 is 25, so √850 = √25 × √34 = 5√34.
What is √850 rounded to two decimal places?
√850 ≈ 29.15 to two decimal places (29.2 to one, 29.155 to three). Check: 29.15² = 849.7225, close to 850.