√855 at a glance
- Exact value
- 3√95
- Decimal (10 places)
- 29.2403830344
- Rounded
- 29.2 · 29.24 · 29.240
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.240383
- Prime factorization
- 3² × 5 × 19
- Cube root
- 9.491220
How to simplify √855
Look for the largest perfect square that divides 855. Here it is 9 (3²), because 855 = 9 × 95 and 95 has no square factor left:
The prime factorization tells the same story: 855 = 3² × 5 × 19. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 5 × 19 stays inside.
Check: (3√95)² = 3² × 95 = 9 × 95 = 855. As a decimal, 3√95 = 3 × 9.7467943448 ≈ 29.2403830344.
Where √855 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √855 lies between 29 and 30. 855 is 14 above 841 and 45 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.2373 (0.01% low)
- Tangent from 29, i.e. 29 + 14 ÷ 58: 29.2414 (0% high)
- Tangent from 30, i.e. 30 − 45 ÷ 60: 29.2500 (0.03% high)
For √855 the tangent at 29 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 855 is just 14 above 841.
Finding √855 with the Babylonian method
If a guess is too big, 855 ÷ 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√855) in one step.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 855 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.4827586207 | 29.2413793103 | 3 |
| 2 | 29.2413793103 | 29.2393867925 | 29.2403830514 | 7 |
| 3 | 29.2403830514 | 29.2403830175 | 29.2403830344 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √855 = 29.2403830344 to every decimal shown.
√855 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √855 the pattern is [29; 4, 6, 4, 58] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √855 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 | 2.4 × 10⁻¹ |
| 117/4 | 29.2500000000 | 9.6 × 10⁻³ |
| 731/25 | 29.2400000000 | 3.8 × 10⁻⁴ |
| 3,041/104 | 29.2403846154 | 1.6 × 10⁻⁶ |
| 177,109/6,057 | 29.2403830279 | 6.5 × 10⁻⁹ |
| 711,477/24,332 | 29.2403830347 | 2.6 × 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 855y² = 1. Its smallest solution in positive whole numbers is x = 3,041, y = 104.
√855 in geometry and everyday measurements
- 855 square feet is 79.4 m². Laid out as a square — a small house footprint or a lot — it is about 29.24 ft (29 ft 3 in) on a side.
- 855 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √855 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √855 as its space diagonal.
- Since √855 = 3√95, a length of √855 is exactly 3 copies of the length √95 laid end to end.
Square roots near √855 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √852 | 2√213 | 29.1890 | No |
| √853 | √853 | 29.2062 | No |
| √854 | √854 | 29.2233 | No |
| √855 | 3√95 | 29.2404 | No |
| √856 | 2√214 | 29.2575 | No |
| √857 | √857 | 29.2746 | No |
| √858 | √858 | 29.2916 | No |
- The cube root of 855 is about 9.491220.
- Squaring undoes the root: (√855)² = 855, while 855² = 731,025 — the number whose square root is 855.
Frequently asked questions
What is the square root of 855?
The square root of 855 is 3√95 in simplest radical form, which is about 29.2403830344. The negative root, −29.240383, also squares to 855.
Is the square root of 855 rational or irrational?
Irrational. 855 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 √855 be simplified?
Yes. The largest perfect square dividing 855 is 9, so √855 = √9 × √95 = 3√95.
What is √855 rounded to two decimal places?
√855 ≈ 29.24 to two decimal places (29.2 to one, 29.240 to three). Check: 29.24² = 854.9776, close to 855.