√582 at a glance
- Exact value
- √582
- Decimal (10 places)
- 24.1246761636
- Rounded
- 24.1 · 24.12 · 24.125
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.124676
- Prime factorization
- 2 × 3 × 97
- Cube root
- 8.349126
How to simplify √582
The prime factorization of 582 is 2 × 3 × 97. Every prime appears only once, so there is no pair to bring outside the radical — √582 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 582, 2, 3 and 97 appear an odd number of times, so √582 is irrational and 24.1246761636 is a rounded value.
Where √582 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √582 lies between 24 and 25. 582 is 6 above 576 and 43 below 625, so the root is closer to 24.
- Straight line between 576 and 625: 24.1224 (0.01% low)
- Tangent from 24, i.e. 24 + 6 ÷ 48: 24.1250 (0% high)
- Tangent from 25, i.e. 25 − 43 ÷ 50: 24.1400 (0.06% high)
For √582 the tangent at 24 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 582 is just 6 above 576.
Finding √582 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 | 582 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 24.2500000000 | 24.1250000000 | 3 |
| 2 | 24.1250000000 | 24.1243523316 | 24.1246761658 | 8 |
| 3 | 24.1246761658 | 24.1246761615 | 24.1246761636 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √582 = 24.1246761636 to every decimal shown.
√582 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √582 the pattern is [24; 8, 48] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √582 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 |
|---|---|---|
| 24/1 | 24.0000000000 | 1.2 × 10⁻¹ |
| 193/8 | 24.1250000000 | 3.2 × 10⁻⁴ |
| 9,288/385 | 24.1246753247 | 8.4 × 10⁻⁷ |
| 74,497/3,088 | 24.1246761658 | 2.2 × 10⁻⁹ |
| 3,585,144/148,609 | 24.1246761636 | < 10⁻¹⁰ |
| 28,755,649/1,191,960 | 24.1246761636 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 582y² = 1. Its smallest solution in positive whole numbers is x = 193, y = 8.
√582 in geometry and everyday measurements
- A square garage floor of 582 square feet measures about 24.12 ft (24 ft 1 in) per side, and its corner-to-corner diagonal is √1164 ≈ 34.1 ft.
- 582 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √582 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 7 × 23 box, because 2² + 7² + 23² = 582.
Square roots near √582 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √579 | √579 | 24.0624 | No |
| √580 | 2√145 | 24.0832 | No |
| √581 | √581 | 24.1039 | No |
| √582 | √582 | 24.1247 | No |
| √583 | √583 | 24.1454 | No |
| √584 | 2√146 | 24.1661 | No |
| √585 | 3√65 | 24.1868 | No |
- The cube root of 582 is about 8.349126.
- Squaring undoes the root: (√582)² = 582, while 582² = 338,724 — the number whose square root is 582.
Frequently asked questions
What is the square root of 582?
The square root of 582 is √582, about 24.1246761636. The negative root, −24.124676, also squares to 582.
Is the square root of 582 rational or irrational?
Irrational. 582 is not a perfect square — it falls between 576 and 625 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √582 be simplified?
No. 582 = 2 × 3 × 97 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √582 rounded to two decimal places?
√582 ≈ 24.12 to two decimal places (24.1 to one, 24.125 to three). Check: 24.12² = 581.7744, close to 582.