√574 at a glance
- Exact value
- √574
- Decimal (10 places)
- 23.9582971014
- Rounded
- 24.0 · 23.96 · 23.958
- Perfect square?
- No — between 23² and 24²
- Rational?
- Irrational
- Both square roots
- ±23.958297
- Prime factorization
- 2 × 7 × 41
- Cube root
- 8.310694
How to simplify √574
The prime factorization of 574 is 2 × 7 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √574 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 574, 2, 7 and 41 appear an odd number of times, so √574 is irrational and 23.9582971014 is a rounded value.
Where √574 sits between perfect squares
529 = 23² and 576 = 24² are the nearest perfect squares, so √574 lies between 23 and 24. 574 is 45 above 529 and 2 below 576, so the root is closer to 24.
- Straight line between 529 and 576: 23.9574 (0% low)
- Tangent from 23, i.e. 23 + 45 ÷ 46: 23.9783 (0.08% high)
- Tangent from 24, i.e. 24 − 2 ÷ 48: 23.9583 (0% high)
For √574 the tangent at 24 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 574 is just 2 below 576.
Finding √574 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 | 574 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 23.9166666667 | 23.9583333333 | 4 |
| 2 | 23.9583333333 | 23.9582608696 | 23.9582971014 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √574 = 23.9582971014 to every decimal shown.
√574 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √574 the pattern is [23; 1, 22, 1, 46] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √574 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 | 9.6 × 10⁻¹ |
| 24/1 | 24.0000000000 | 4.2 × 10⁻² |
| 551/23 | 23.9565217391 | 1.8 × 10⁻³ |
| 575/24 | 23.9583333333 | 3.6 × 10⁻⁵ |
| 27,001/1,127 | 23.9582963620 | 7.4 × 10⁻⁷ |
| 27,576/1,151 | 23.9582971329 | 3.2 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 574y² = 1. Its smallest solution in positive whole numbers is x = 575, y = 24.
√574 in geometry and everyday measurements
- A square garage floor of 574 square feet measures about 23.96 ft (23 ft 11 in) per side, and its corner-to-corner diagonal is √1148 ≈ 33.9 ft.
- 574 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √574 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 6 × 23 box, because 3² + 6² + 23² = 574.
Square roots near √574 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √571 | √571 | 23.8956 | No |
| √572 | 2√143 | 23.9165 | No |
| √573 | √573 | 23.9374 | No |
| √574 | √574 | 23.9583 | No |
| √575 | 5√23 | 23.9792 | No |
| √576 | 24 | 24.0000 | Yes |
| √577 | √577 | 24.0208 | No |
- The cube root of 574 is about 8.310694.
- Squaring undoes the root: (√574)² = 574, while 574² = 329,476 — the number whose square root is 574.
Frequently asked questions
What is the square root of 574?
The square root of 574 is √574, about 23.9582971014. The negative root, −23.958297, also squares to 574.
Is the square root of 574 rational or irrational?
Irrational. 574 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 √574 be simplified?
No. 574 = 2 × 7 × 41 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √574 rounded to two decimal places?
√574 ≈ 23.96 to two decimal places (24.0 to one, 23.958 to three). Check: 23.96² = 574.0816, close to 574.