√810 at a glance
- Exact value
- 9√10
- Decimal (10 places)
- 28.4604989415
- Rounded
- 28.5 · 28.46 · 28.460
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.460499
- Prime factorization
- 2 × 3⁴ × 5
- Cube root
- 9.321698
How to simplify √810
Look for the largest perfect square that divides 810. Here it is 81 (9²), because 810 = 81 × 10 and 10 has no square factor left:
The prime factorization tells the same story: 810 = 2 × 3⁴ × 5. Each pair of equal primes leaves the radical as one factor, so 3² comes out and 2 × 5 stays inside.
810 has 2 square factors (9 and 81). Starting with a smaller one still works but takes more rounds: √810 = 3√90, and √90 can be simplified again. Using 81 straight away finishes in one step.
Check: (9√10)² = 9² × 10 = 81 × 10 = 810. As a decimal, 9√10 = 9 × 3.1622776602 ≈ 28.4604989415.
Where √810 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √810 lies between 28 and 29. 810 is 26 above 784 and 31 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.4561 (0.02% low)
- Tangent from 28, i.e. 28 + 26 ÷ 56: 28.4643 (0.01% high)
- Tangent from 29, i.e. 29 − 31 ÷ 58: 28.4655 (0.02% high)
For √810 the tangent at 28 wins, missing by only 0.0038. Tangent estimates shine when the number sits close to a perfect square — here 810 is just 26 above 784.
Finding √810 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, 28 (28² = 784):
| Step | Guess x | 810 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.9285714286 | 28.4642857143 | 2 |
| 2 | 28.4642857143 | 28.4567126725 | 28.4604991934 | 6 |
| 3 | 28.4604991934 | 28.4604986896 | 28.4604989415 | 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 √810 = 28.4604989415 to every decimal shown.
√810 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √810 the pattern is [28; 2, 5, 1, 4, 1, 5, 2, 56] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √810 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 | 4.6 × 10⁻¹ |
| 57/2 | 28.5000000000 | 4.0 × 10⁻² |
| 313/11 | 28.4545454545 | 6.0 × 10⁻³ |
| 370/13 | 28.4615384615 | 1.0 × 10⁻³ |
| 1,793/63 | 28.4603174603 | 1.8 × 10⁻⁴ |
| 2,163/76 | 28.4605263158 | 2.7 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 810y² = 1. Its smallest solution in positive whole numbers is x = 27,379, y = 962.
√810 in geometry and everyday measurements
- 810 square feet is 75.3 m². Laid out as a square — a small house footprint or a lot — it is about 28.46 ft (28 ft 6 in) on a side.
- 810 = 9² + 27², so by the Pythagorean theorem √810 is the diagonal of a 9 × 27 rectangle — and the distance between the points (0, 0) and (9, 27) on a grid.
- Since √810 = 9√10, a length of √810 is exactly 9 copies of the length √10 laid end to end.
Square roots near √810 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √807 | √807 | 28.4077 | No |
| √808 | 2√202 | 28.4253 | No |
| √809 | √809 | 28.4429 | No |
| √810 | 9√10 | 28.4605 | No |
| √811 | √811 | 28.4781 | No |
| √812 | 2√203 | 28.4956 | No |
| √813 | √813 | 28.5132 | No |
- The cube root of 810 is about 9.321698.
- Squaring undoes the root: (√810)² = 810, while 810² = 656,100 — the number whose square root is 810.
Frequently asked questions
What is the square root of 810?
The square root of 810 is 9√10 in simplest radical form, which is about 28.4604989415. The negative root, −28.460499, also squares to 810.
Is the square root of 810 rational or irrational?
Irrational. 810 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 √810 be simplified?
Yes. The largest perfect square dividing 810 is 81, so √810 = √81 × √10 = 9√10.
What is √810 rounded to two decimal places?
√810 ≈ 28.46 to two decimal places (28.5 to one, 28.460 to three). Check: 28.46² = 809.9716, close to 810.