√869 at a glance
- Exact value
- √869
- Decimal (10 places)
- 29.4788059460
- Rounded
- 29.5 · 29.48 · 29.479
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.478806
- Prime factorization
- 11 × 79
- Cube root
- 9.542744
How to simplify √869
The prime factorization of 869 is 11 × 79. Every prime appears only once, so there is no pair to bring outside the radical — √869 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 869, 11 and 79 appear an odd number of times, so √869 is irrational and 29.4788059460 is a rounded value.
Where √869 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √869 lies between 29 and 30. 869 is 28 above 841 and 31 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.4746 (0.01% low)
- Tangent from 29, i.e. 29 + 28 ÷ 58: 29.4828 (0.01% high)
- Tangent from 30, i.e. 30 − 31 ÷ 60: 29.4833 (0.02% high)
For √869 the tangent at 29 wins, missing by only 0.004. Tangent estimates shine when the number sits close to a perfect square — here 869 is just 28 above 841.
Finding √869 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 869: following the tangent line down to zero simplifies to averaging x with 869 ÷ x.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 869 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.9655172414 | 29.4827586207 | 2 |
| 2 | 29.4827586207 | 29.4748538012 | 29.4788062109 | 6 |
| 3 | 29.4788062109 | 29.4788056810 | 29.4788059460 | 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 √869 = 29.4788059460 to every decimal shown.
√869 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √869 the pattern is [29; 2, 11, 3, 2, 1, 1, 1, 1, 1, 14, 8, 2, …] with the block of 28 terms after the semicolon repeating forever (only the first 12 of the 28 are shown). A pattern that never ends is one more proof that √869 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 | 4.8 × 10⁻¹ |
| 59/2 | 29.5000000000 | 2.1 × 10⁻² |
| 678/23 | 29.4782608696 | 5.5 × 10⁻⁴ |
| 2,093/71 | 29.4788732394 | 6.7 × 10⁻⁵ |
| 4,864/165 | 29.4787878788 | 1.8 × 10⁻⁵ |
| 6,957/236 | 29.4788135593 | 7.6 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 869y² = 1. Its smallest solution in positive whole numbers is x = 60,192,738,698,751, y = 2,041,898,807,200 — 14 digits for x, even though 869 is small, which is what makes Pell’s equation famous.
√869 in geometry and everyday measurements
- 869 square feet is 80.7 m². Laid out as a square — a small house footprint or a lot — it is about 29.48 ft (29 ft 6 in) on a side.
- 869 is not a sum of two whole-number squares — the prime factor 11 and 79 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √869 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 9 × 28 box, because 2² + 9² + 28² = 869.
Square roots near √869 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √866 | √866 | 29.4279 | No |
| √867 | 17√3 | 29.4449 | No |
| √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 |
- The cube root of 869 is about 9.542744.
- Squaring undoes the root: (√869)² = 869, while 869² = 755,161 — the number whose square root is 869.
Frequently asked questions
What is the square root of 869?
The square root of 869 is √869, about 29.4788059460. The negative root, −29.478806, also squares to 869.
Is the square root of 869 rational or irrational?
Irrational. 869 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 √869 be simplified?
No. 869 = 11 × 79 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √869 rounded to two decimal places?
√869 ≈ 29.48 to two decimal places (29.5 to one, 29.479 to three). Check: 29.48² = 869.0704, close to 869.