√851 at a glance
- Exact value
- √851
- Decimal (10 places)
- 29.1719042916
- Rounded
- 29.2 · 29.17 · 29.172
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.171904
- Prime factorization
- 23 × 37
- Cube root
- 9.476396
How to simplify √851
The prime factorization of 851 is 23 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √851 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 851, 23 and 37 appear an odd number of times, so √851 is irrational and 29.1719042916 is a rounded value.
Where √851 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √851 lies between 29 and 30. 851 is 10 above 841 and 49 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.1695 (0.01% low)
- Tangent from 29, i.e. 29 + 10 ÷ 58: 29.1724 (0% high)
- Tangent from 30, i.e. 30 − 49 ÷ 60: 29.1833 (0.04% high)
For √851 the tangent at 29 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 851 is just 10 above 841.
Finding √851 with the Babylonian method
If a guess is too big, 851 ÷ 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√851) in one step.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 851 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.3448275862 | 29.1724137931 | 3 |
| 2 | 29.1724137931 | 29.1713947991 | 29.1719042961 | 8 |
| 3 | 29.1719042961 | 29.1719042872 | 29.1719042916 | 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 √851 = 29.1719042916 to every decimal shown.
√851 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √851 the pattern is [29; 5, 1, 4, 2, 7, 1, 7, 2, 4, 1, 5, 58] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √851 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 | 1.7 × 10⁻¹ |
| 146/5 | 29.2000000000 | 2.8 × 10⁻² |
| 175/6 | 29.1666666667 | 5.2 × 10⁻³ |
| 846/29 | 29.1724137931 | 5.1 × 10⁻⁴ |
| 1,867/64 | 29.1718750000 | 2.9 × 10⁻⁵ |
| 13,915/477 | 29.1719077568 | 3.5 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 851y² = 1. Its smallest solution in positive whole numbers is x = 8,418,574, y = 288,585.
√851 in geometry and everyday measurements
- 851 square feet is 79.1 m². Laid out as a square — a small house footprint or a lot — it is about 29.17 ft (29 ft 2 in) on a side.
- 851 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √851 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 29 box, because 1² + 3² + 29² = 851.
Square roots near √851 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √854 | √854 | 29.2233 | No |
- The cube root of 851 is about 9.476396.
- Squaring undoes the root: (√851)² = 851, while 851² = 724,201 — the number whose square root is 851.
Frequently asked questions
What is the square root of 851?
The square root of 851 is √851, about 29.1719042916. The negative root, −29.171904, also squares to 851.
Is the square root of 851 rational or irrational?
Irrational. 851 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 √851 be simplified?
No. 851 = 23 × 37 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √851 rounded to two decimal places?
√851 ≈ 29.17 to two decimal places (29.2 to one, 29.172 to three). Check: 29.17² = 850.8889, close to 851.