√818 at a glance
- Exact value
- √818
- Decimal (10 places)
- 28.6006992922
- Rounded
- 28.6 · 28.60 · 28.601
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.600699
- Prime factorization
- 2 × 409
- Cube root
- 9.352286
How to simplify √818
The prime factorization of 818 is 2 × 409. Every prime appears only once, so there is no pair to bring outside the radical — √818 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 818, 2 and 409 appear an odd number of times, so √818 is irrational and 28.6006992922 is a rounded value.
Where √818 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √818 lies between 28 and 29. 818 is 34 above 784 and 23 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.5965 (0.01% low)
- Tangent from 28, i.e. 28 + 34 ÷ 56: 28.6071 (0.02% high)
- Tangent from 29, i.e. 29 − 23 ÷ 58: 28.6034 (0.01% high)
For √818 the tangent at 29 wins, missing by only 0.0027. Tangent estimates shine when the number sits close to a perfect square — here 818 is just 23 below 841.
Finding √818 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 | 818 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.2068965517 | 28.6034482759 | 2 |
| 2 | 28.6034482759 | 28.5979505726 | 28.6006994242 | 6 |
| 3 | 28.6006994242 | 28.6006991601 | 28.6006992922 | 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 √818 = 28.6006992922 to every decimal shown.
√818 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √818 the pattern is [28; 1, 1, 1, 1, 56] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √818 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 | 6.0 × 10⁻¹ |
| 29/1 | 29.0000000000 | 4.0 × 10⁻¹ |
| 57/2 | 28.5000000000 | 1.0 × 10⁻¹ |
| 86/3 | 28.6666666667 | 6.6 × 10⁻² |
| 143/5 | 28.6000000000 | 7.0 × 10⁻⁴ |
| 8,094/283 | 28.6007067138 | 7.4 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 818y² = 1. Its smallest solution in positive whole numbers is x = 40,899, y = 1,430. Because the period is odd, the equation with −1 on the right also has a solution: 143² − 818 × 5² = −1.
√818 in geometry and everyday measurements
- 818 square feet is 76 m². Laid out as a square — a small house footprint or a lot — it is about 28.6 ft (28 ft 7 in) on a side.
- 818 = 17² + 23², so by the Pythagorean theorem √818 is the diagonal of a 17 × 23 rectangle — and the distance between the points (0, 0) and (17, 23) on a grid.
Square roots near √818 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √815 | √815 | 28.5482 | No |
| √816 | 4√51 | 28.5657 | No |
| √817 | √817 | 28.5832 | No |
| √818 | √818 | 28.6007 | No |
| √819 | 3√91 | 28.6182 | No |
| √820 | 2√205 | 28.6356 | No |
| √821 | √821 | 28.6531 | No |
- The cube root of 818 is about 9.352286.
- Squaring undoes the root: (√818)² = 818, while 818² = 669,124 — the number whose square root is 818.
Frequently asked questions
What is the square root of 818?
The square root of 818 is √818, about 28.6006992922. The negative root, −28.600699, also squares to 818.
Is the square root of 818 rational or irrational?
Irrational. 818 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 √818 be simplified?
No. 818 = 2 × 409 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √818 rounded to two decimal places?
√818 ≈ 28.60 to two decimal places (28.6 to one, 28.601 to three). Check: 28.60² = 817.96, close to 818.