√838 at a glance
- Exact value
- √838
- Decimal (10 places)
- 28.9482296523
- Rounded
- 28.9 · 28.95 · 28.948
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.948230
- Prime factorization
- 2 × 419
- Cube root
- 9.427894
How to simplify √838
The prime factorization of 838 is 2 × 419. Every prime appears only once, so there is no pair to bring outside the radical — √838 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 838, 2 and 419 appear an odd number of times, so √838 is irrational and 28.9482296523 is a rounded value.
Where √838 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √838 lies between 28 and 29. 838 is 54 above 784 and 3 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.9474 (0% low)
- Tangent from 28, i.e. 28 + 54 ÷ 56: 28.9643 (0.06% high)
- Tangent from 29, i.e. 29 − 3 ÷ 58: 28.9483 (0% high)
For √838 the tangent at 29 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 838 is just 3 below 841.
Finding √838 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 838 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.8965517241 | 28.9482758621 | 4 |
| 2 | 28.9482758621 | 28.9481834425 | 28.9482296523 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √838 = 28.9482296523 to every decimal shown.
√838 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √838 the pattern is [28; 1, 18, 3, 6, 9, 2, 28, 2, 9, 6, 3, 18, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √838 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 | 9.5 × 10⁻¹ |
| 29/1 | 29.0000000000 | 5.2 × 10⁻² |
| 550/19 | 28.9473684211 | 8.6 × 10⁻⁴ |
| 1,679/58 | 28.9482758621 | 4.6 × 10⁻⁵ |
| 10,624/367 | 28.9482288828 | 7.7 × 10⁻⁷ |
| 97,295/3,361 | 28.9482296935 | 4.1 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 838y² = 1. Its smallest solution in positive whole numbers is x = 42,112,785,797, y = 1,454,762,046.
√838 in geometry and everyday measurements
- 838 square feet is 77.9 m². Laid out as a square — a small house footprint or a lot — it is about 28.95 ft (28 ft 11 in) on a side.
- 838 is not a sum of two whole-number squares — the prime factor 419 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √838 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 10 × 27 box, because 3² + 10² + 27² = 838.
Square roots near √838 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √835 | √835 | 28.8964 | No |
| √836 | 2√209 | 28.9137 | No |
| √837 | 3√93 | 28.9310 | No |
| √838 | √838 | 28.9482 | No |
| √839 | √839 | 28.9655 | No |
| √840 | 2√210 | 28.9828 | No |
| √841 | 29 | 29.0000 | Yes |
- The cube root of 838 is about 9.427894.
- Squaring undoes the root: (√838)² = 838, while 838² = 702,244 — the number whose square root is 838.
Frequently asked questions
What is the square root of 838?
The square root of 838 is √838, about 28.9482296523. The negative root, −28.948230, also squares to 838.
Is the square root of 838 rational or irrational?
Irrational. 838 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 √838 be simplified?
No. 838 = 2 × 419 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √838 rounded to two decimal places?
√838 ≈ 28.95 to two decimal places (28.9 to one, 28.948 to three). Check: 28.95² = 838.1025, close to 838.