√817 at a glance
- Exact value
- √817
- Decimal (10 places)
- 28.5832118559
- Rounded
- 28.6 · 28.58 · 28.583
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.583212
- Prime factorization
- 19 × 43
- Cube root
- 9.348473
How to simplify √817
The prime factorization of 817 is 19 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √817 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 817, 19 and 43 appear an odd number of times, so √817 is irrational and 28.5832118559 is a rounded value.
Where √817 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √817 lies between 28 and 29. 817 is 33 above 784 and 24 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.5789 (0.01% low)
- Tangent from 28, i.e. 28 + 33 ÷ 56: 28.5893 (0.02% high)
- Tangent from 29, i.e. 29 − 24 ÷ 58: 28.5862 (0.01% high)
For √817 the tangent at 29 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 817 is just 24 below 841.
Finding √817 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 817: following the tangent line down to zero simplifies to averaging x with 817 ÷ x.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 817 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.1724137931 | 28.5862068966 | 2 |
| 2 | 28.5862068966 | 28.5802171291 | 28.5832120128 | 6 |
| 3 | 28.5832120128 | 28.5832116990 | 28.5832118559 | 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 √817 = 28.5832118559 to every decimal shown.
√817 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √817 the pattern is [28; 1, 1, 2, 1, 1, 56] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √817 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 | 5.8 × 10⁻¹ |
| 29/1 | 29.0000000000 | 4.2 × 10⁻¹ |
| 57/2 | 28.5000000000 | 8.3 × 10⁻² |
| 143/5 | 28.6000000000 | 1.7 × 10⁻² |
| 200/7 | 28.5714285714 | 1.2 × 10⁻² |
| 343/12 | 28.5833333333 | 1.2 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 817y² = 1. Its smallest solution in positive whole numbers is x = 343, y = 12.
√817 in geometry and everyday measurements
- 817 square feet is 75.9 m². Laid out as a square — a small house footprint or a lot — it is about 28.58 ft (28 ft 7 in) on a side.
- 817 is not a sum of two whole-number squares — the prime factor 19 and 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √817 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 18 × 22 box, because 3² + 18² + 22² = 817.
Square roots near √817 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √814 | √814 | 28.5307 | No |
| √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 |
- The cube root of 817 is about 9.348473.
- Squaring undoes the root: (√817)² = 817, while 817² = 667,489 — the number whose square root is 817.
Frequently asked questions
What is the square root of 817?
The square root of 817 is √817, about 28.5832118559. The negative root, −28.583212, also squares to 817.
Is the square root of 817 rational or irrational?
Irrational. 817 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 √817 be simplified?
No. 817 = 19 × 43 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √817 rounded to two decimal places?
√817 ≈ 28.58 to two decimal places (28.6 to one, 28.583 to three). Check: 28.58² = 816.8164, close to 817.