√578 at a glance
- Exact value
- 17√2
- Decimal (10 places)
- 24.0416305603
- Rounded
- 24.0 · 24.04 · 24.042
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.041631
- Prime factorization
- 2 × 17²
- Cube root
- 8.329954
How to simplify √578
Look for the largest perfect square that divides 578. Here it is 289 (17²), because 578 = 289 × 2 and 2 has no square factor left:
The prime factorization tells the same story: 578 = 2 × 17². Each pair of equal primes leaves the radical as one factor, so 17 comes out and 2 stays inside.
Check: (17√2)² = 17² × 2 = 289 × 2 = 578. As a decimal, 17√2 = 17 × 1.4142135624 ≈ 24.0416305603.
Where √578 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √578 lies between 24 and 25. 578 is 2 above 576 and 47 below 625, so the root is closer to 24.
- Straight line between 576 and 625: 24.0408 (0% low)
- Tangent from 24, i.e. 24 + 2 ÷ 48: 24.0417 (0% high)
- Tangent from 25, i.e. 25 − 47 ÷ 50: 24.0600 (0.08% high)
For √578 the tangent at 24 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 578 is just 2 above 576.
Finding √578 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 | 578 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 24.0833333333 | 24.0416666667 | 4 |
| 2 | 24.0416666667 | 24.0415944541 | 24.0416305604 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √578 = 24.0416305603 to every decimal shown.
√578 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √578 the pattern is [24; 24, 48] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √578 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 | 4.2 × 10⁻² |
| 577/24 | 24.0416666667 | 3.6 × 10⁻⁵ |
| 27,720/1,153 | 24.0416305291 | 3.1 × 10⁻⁸ |
| 665,857/27,696 | 24.0416305604 | < 10⁻¹⁰ |
| 31,988,856/1,330,561 | 24.0416305603 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 578y² = 1. Its smallest solution in positive whole numbers is x = 577, y = 24.
√578 in geometry and everyday measurements
- A square garage floor of 578 square feet measures about 24.04 ft (24 ft) per side, and its corner-to-corner diagonal is √1156 ≈ 34 ft.
- 578 = 7² + 23² = 17² + 17², so by the Pythagorean theorem √578 is the diagonal of rectangles measuring 7 × 23 and 17 × 17 — and the distance between the points (0, 0) and (7, 23) on a grid.
- Since √578 = 17√2, a length of √578 is exactly 17 copies of the length √2 laid end to end.
Square roots near √578 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √575 | 5√23 | 23.9792 | No |
| √576 | 24 | 24.0000 | Yes |
| √577 | √577 | 24.0208 | No |
| √578 | 17√2 | 24.0416 | No |
| √579 | √579 | 24.0624 | No |
| √580 | 2√145 | 24.0832 | No |
| √581 | √581 | 24.1039 | No |
- The cube root of 578 is about 8.329954.
- Squaring undoes the root: (√578)² = 578, while 578² = 334,084 — the number whose square root is 578.
Frequently asked questions
What is the square root of 578?
The square root of 578 is 17√2 in simplest radical form, which is about 24.0416305603. The negative root, −24.041631, also squares to 578.
Is the square root of 578 rational or irrational?
Irrational. 578 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 √578 be simplified?
Yes. The largest perfect square dividing 578 is 289, so √578 = √289 × √2 = 17√2.
What is √578 rounded to two decimal places?
√578 ≈ 24.04 to two decimal places (24.0 to one, 24.042 to three). Check: 24.04² = 577.9216, close to 578.