√813 at a glance
- Exact value
- √813
- Decimal (10 places)
- 28.5131548588
- Rounded
- 28.5 · 28.51 · 28.513
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.513155
- Prime factorization
- 3 × 271
- Cube root
- 9.333192
How to simplify √813
The prime factorization of 813 is 3 × 271. Every prime appears only once, so there is no pair to bring outside the radical — √813 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 813, 3 and 271 appear an odd number of times, so √813 is irrational and 28.5131548588 is a rounded value.
Where √813 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √813 lies between 28 and 29. 813 is 29 above 784 and 28 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.5088 (0.02% low)
- Tangent from 28, i.e. 28 + 29 ÷ 56: 28.5179 (0.02% high)
- Tangent from 29, i.e. 29 − 28 ÷ 58: 28.5172 (0.01% high)
For √813 the tangent at 29 wins, missing by only 0.0041. Tangent estimates shine when the number sits close to a perfect square — here 813 is just 28 below 841.
Finding √813 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 813: following the tangent line down to zero simplifies to averaging x with 813 ÷ x.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 813 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.0344827586 | 28.5172413793 | 2 |
| 2 | 28.5172413793 | 28.5090689238 | 28.5131551516 | 6 |
| 3 | 28.5131551516 | 28.5131545660 | 28.5131548588 | 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 √813 = 28.5131548588 to every decimal shown.
√813 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √813 the pattern is [28; 1, 1, 18, 1, 1, 56] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √813 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 | 5.1 × 10⁻¹ |
| 29/1 | 29.0000000000 | 4.9 × 10⁻¹ |
| 57/2 | 28.5000000000 | 1.3 × 10⁻² |
| 1,055/37 | 28.5135135135 | 3.6 × 10⁻⁴ |
| 1,112/39 | 28.5128205128 | 3.3 × 10⁻⁴ |
| 2,167/76 | 28.5131578947 | 3.0 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 813y² = 1. Its smallest solution in positive whole numbers is x = 2,167, y = 76.
√813 in geometry and everyday measurements
- 813 square feet is 75.5 m². Laid out as a square — a small house footprint or a lot — it is about 28.51 ft (28 ft 6 in) on a side.
- 813 is not a sum of two whole-number squares — the prime factor 3 and 271 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √813 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 5 × 28 box, because 2² + 5² + 28² = 813.
Square roots near √813 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √810 | 9√10 | 28.4605 | No |
| √811 | √811 | 28.4781 | No |
| √812 | 2√203 | 28.4956 | No |
| √813 | √813 | 28.5132 | No |
| √814 | √814 | 28.5307 | No |
| √815 | √815 | 28.5482 | No |
| √816 | 4√51 | 28.5657 | No |
- The cube root of 813 is about 9.333192.
- Squaring undoes the root: (√813)² = 813, while 813² = 660,969 — the number whose square root is 813.
Frequently asked questions
What is the square root of 813?
The square root of 813 is √813, about 28.5131548588. The negative root, −28.513155, also squares to 813.
Is the square root of 813 rational or irrational?
Irrational. 813 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 √813 be simplified?
No. 813 = 3 × 271 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √813 rounded to two decimal places?
√813 ≈ 28.51 to two decimal places (28.5 to one, 28.513 to three). Check: 28.51² = 812.8201, close to 813.