√566 at a glance
- Exact value
- √566
- Decimal (10 places)
- 23.7907545067
- Rounded
- 23.8 · 23.79 · 23.791
- Perfect square?
- No — between 23² and 24²
- Rational?
- Irrational
- Both square roots
- ±23.790755
- Prime factorization
- 2 × 283
- Cube root
- 8.271904
How to simplify √566
The prime factorization of 566 is 2 × 283. Every prime appears only once, so there is no pair to bring outside the radical — √566 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 566, 2 and 283 appear an odd number of times, so √566 is irrational and 23.7907545067 is a rounded value.
Where √566 sits between perfect squares
529 = 23² and 576 = 24² are the nearest perfect squares, so √566 lies between 23 and 24. 566 is 37 above 529 and 10 below 576, so the root is closer to 24.
- Straight line between 529 and 576: 23.7872 (0.01% low)
- Tangent from 23, i.e. 23 + 37 ÷ 46: 23.8043 (0.06% high)
- Tangent from 24, i.e. 24 − 10 ÷ 48: 23.7917 (0% high)
For √566 the tangent at 24 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 566 is just 10 below 576.
Finding √566 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, 24 (24² = 576):
| Step | Guess x | 566 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 23.5833333333 | 23.7916666667 | 3 |
| 2 | 23.7916666667 | 23.7898423818 | 23.7907545242 | 7 |
| 3 | 23.7907545242 | 23.7907544893 | 23.7907545067 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √566 = 23.7907545067 to every decimal shown.
√566 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √566 the pattern is [23; 1, 3, 1, 3, 1, 1, 8, 1, 22, 1, 8, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √566 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 |
|---|---|---|
| 23/1 | 23.0000000000 | 7.9 × 10⁻¹ |
| 24/1 | 24.0000000000 | 2.1 × 10⁻¹ |
| 95/4 | 23.7500000000 | 4.1 × 10⁻² |
| 119/5 | 23.8000000000 | 9.2 × 10⁻³ |
| 452/19 | 23.7894736842 | 1.3 × 10⁻³ |
| 571/24 | 23.7916666667 | 9.1 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 566y² = 1. Its smallest solution in positive whole numbers is x = 95,609,285, y = 4,018,758.
√566 in geometry and everyday measurements
- A square garage floor of 566 square feet measures about 23.79 ft (23 ft 9 in) per side, and its corner-to-corner diagonal is √1132 ≈ 33.6 ft.
- 566 is not a sum of two whole-number squares — the prime factor 283 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √566 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 23 box, because 1² + 6² + 23² = 566.
Square roots near √566 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √563 | √563 | 23.7276 | No |
| √564 | 2√141 | 23.7487 | No |
| √565 | √565 | 23.7697 | No |
| √566 | √566 | 23.7908 | No |
| √567 | 9√7 | 23.8118 | No |
| √568 | 2√142 | 23.8328 | No |
| √569 | √569 | 23.8537 | No |
- The cube root of 566 is about 8.271904.
- Squaring undoes the root: (√566)² = 566, while 566² = 320,356 — the number whose square root is 566.
Frequently asked questions
What is the square root of 566?
The square root of 566 is √566, about 23.7907545067. The negative root, −23.790755, also squares to 566.
Is the square root of 566 rational or irrational?
Irrational. 566 is not a perfect square — it falls between 529 and 576 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √566 be simplified?
No. 566 = 2 × 283 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √566 rounded to two decimal places?
√566 ≈ 23.79 to two decimal places (23.8 to one, 23.791 to three). Check: 23.79² = 565.9641, close to 566.