√871 at a glance
- Exact value
- √871
- Decimal (10 places)
- 29.5127091267
- Rounded
- 29.5 · 29.51 · 29.513
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.512709
- Prime factorization
- 13 × 67
- Cube root
- 9.550059
How to simplify √871
The prime factorization of 871 is 13 × 67. Every prime appears only once, so there is no pair to bring outside the radical — √871 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 871, 13 and 67 appear an odd number of times, so √871 is irrational and 29.5127091267 is a rounded value.
Where √871 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √871 lies between 29 and 30. 871 is 30 above 841 and 29 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.5085 (0.01% low)
- Tangent from 29, i.e. 29 + 30 ÷ 58: 29.5172 (0.02% high)
- Tangent from 30, i.e. 30 − 29 ÷ 60: 29.5167 (0.01% high)
For √871 the tangent at 30 wins, missing by only 0.004. Tangent estimates shine when the number sits close to a perfect square — here 871 is just 29 below 900.
Finding √871 with the Babylonian method
If a guess is too big, 871 ÷ 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√871) in one step.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 871 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.0333333333 | 29.5166666667 | 2 |
| 2 | 29.5166666667 | 29.5087521174 | 29.5127093921 | 6 |
| 3 | 29.5127093921 | 29.5127088614 | 29.5127091267 | 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 √871 = 29.5127091267 to every decimal shown.
√871 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √871 the pattern is [29; 1, 1, 19, 5, 1, 5, 1, 2, 1, 1, 1, 1, …] with the block of 24 terms after the semicolon repeating forever (only the first 12 of the 24 are shown). A pattern that never ends is one more proof that √871 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 | 5.1 × 10⁻¹ |
| 30/1 | 30.0000000000 | 4.9 × 10⁻¹ |
| 59/2 | 29.5000000000 | 1.3 × 10⁻² |
| 1,151/39 | 29.5128205128 | 1.1 × 10⁻⁴ |
| 5,814/197 | 29.5126903553 | 1.9 × 10⁻⁵ |
| 6,965/236 | 29.5127118644 | 2.7 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 871y² = 1. Its smallest solution in positive whole numbers is x = 19,442,812,076, y = 658,794,555.
√871 in geometry and everyday measurements
- 871 square feet is 80.9 m². Laid out as a square — a small house footprint or a lot — it is about 29.51 ft (29 ft 6 in) on a side.
- 871 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 √871 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 √871 as its space diagonal.
Square roots near √871 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √868 | 2√217 | 29.4618 | No |
| √869 | √869 | 29.4788 | No |
| √870 | √870 | 29.4958 | No |
| √871 | √871 | 29.5127 | No |
| √872 | 2√218 | 29.5296 | No |
| √873 | 3√97 | 29.5466 | No |
| √874 | √874 | 29.5635 | No |
- The cube root of 871 is about 9.550059.
- Squaring undoes the root: (√871)² = 871, while 871² = 758,641 — the number whose square root is 871.
Frequently asked questions
What is the square root of 871?
The square root of 871 is √871, about 29.5127091267. The negative root, −29.512709, also squares to 871.
Is the square root of 871 rational or irrational?
Irrational. 871 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 √871 be simplified?
No. 871 = 13 × 67 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √871 rounded to two decimal places?
√871 ≈ 29.51 to two decimal places (29.5 to one, 29.513 to three). Check: 29.51² = 870.8401, close to 871.