√829 at a glance
- Exact value
- √829
- Decimal (10 places)
- 28.7923600978
- Rounded
- 28.8 · 28.79 · 28.792
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.792360
- Prime factorization
- 829
- Cube root
- 9.394021
How to simplify √829
829 is a prime number, so its only factors are 1 and 829. There is no perfect-square factor to pull out, which means √829 is already in its simplest radical form.
The square root of any prime is irrational. If √829 were a fraction a/b in lowest terms, then a² = 829b², so 829 would divide a — and then 829 would divide b too, contradicting “lowest terms.” That is why the decimal 28.7923600978 is only a rounded value.
Where √829 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √829 lies between 28 and 29. 829 is 45 above 784 and 12 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.7895 (0.01% low)
- Tangent from 28, i.e. 28 + 45 ÷ 56: 28.8036 (0.04% high)
- Tangent from 29, i.e. 29 − 12 ÷ 58: 28.7931 (0% high)
For √829 the tangent at 29 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 829 is just 12 below 841.
Finding √829 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 829: following the tangent line down to zero simplifies to averaging x with 829 ÷ x.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 829 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.5862068966 | 28.7931034483 | 3 |
| 2 | 28.7931034483 | 28.7916167665 | 28.7923601074 | 8 |
| 3 | 28.7923601074 | 28.7923600882 | 28.7923600978 | 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 √829 = 28.7923600978 to every decimal shown.
√829 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √829 the pattern is [28; 1, 3, 1, 4, 2, 3, 2, 1, 1, 2, 3, 2, …] with the block of 17 terms after the semicolon repeating forever (only the first 12 of the 17 are shown). A pattern that never ends is one more proof that √829 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 |
|---|---|---|
| 28/1 | 28.0000000000 | 7.9 × 10⁻¹ |
| 29/1 | 29.0000000000 | 2.1 × 10⁻¹ |
| 115/4 | 28.7500000000 | 4.2 × 10⁻² |
| 144/5 | 28.8000000000 | 7.6 × 10⁻³ |
| 691/24 | 28.7916666667 | 6.9 × 10⁻⁴ |
| 1,526/53 | 28.7924528302 | 9.3 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 829y² = 1. Its smallest solution in positive whole numbers is x = 479,835,713,751,049, y = 16,665,383,182,260 — 15 digits for x, even though 829 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 15,489,282² − 829 × 537,965² = −1.
√829 in geometry and everyday measurements
- 829 square feet is 77 m². Laid out as a square — a small house footprint or a lot — it is about 28.79 ft (28 ft 10 in) on a side.
- 829 = 10² + 27², so by the Pythagorean theorem √829 is the diagonal of a 10 × 27 rectangle — and the distance between the points (0, 0) and (10, 27) on a grid.
Square roots near √829 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √826 | √826 | 28.7402 | No |
| √827 | √827 | 28.7576 | No |
| √828 | 6√23 | 28.7750 | No |
| √829 | √829 | 28.7924 | No |
| √830 | √830 | 28.8097 | No |
| √831 | √831 | 28.8271 | No |
| √832 | 8√13 | 28.8444 | No |
- The cube root of 829 is about 9.394021.
- Squaring undoes the root: (√829)² = 829, while 829² = 687,241 — the number whose square root is 829.
Frequently asked questions
What is the square root of 829?
The square root of 829 is √829, about 28.7923600978. The negative root, −28.792360, also squares to 829.
Is the square root of 829 rational or irrational?
Irrational. 829 is not a perfect square — it falls between 784 and 841 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √829 be simplified?
No. 829 is prime, so there is no perfect square to take out of the radical.
What is √829 rounded to two decimal places?
√829 ≈ 28.79 to two decimal places (28.8 to one, 28.792 to three). Check: 28.79² = 828.8641, close to 829.