√573 at a glance
- Exact value
- √573
- Decimal (10 places)
- 23.9374184072
- Rounded
- 23.9 · 23.94 · 23.937
- Perfect square?
- No — between 23² and 24²
- Rational?
- Irrational
- Both square roots
- ±23.937418
- Prime factorization
- 3 × 191
- Cube root
- 8.305865
How to simplify √573
The prime factorization of 573 is 3 × 191. Every prime appears only once, so there is no pair to bring outside the radical — √573 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 573, 3 and 191 appear an odd number of times, so √573 is irrational and 23.9374184072 is a rounded value.
Where √573 sits between perfect squares
529 = 23² and 576 = 24² are the nearest perfect squares, so √573 lies between 23 and 24. 573 is 44 above 529 and 3 below 576, so the root is closer to 24.
- Straight line between 529 and 576: 23.9362 (0.01% low)
- Tangent from 23, i.e. 23 + 44 ÷ 46: 23.9565 (0.08% high)
- Tangent from 24, i.e. 24 − 3 ÷ 48: 23.9375 (0% high)
For √573 the tangent at 24 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 573 is just 3 below 576.
Finding √573 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 573: following the tangent line down to zero simplifies to averaging x with 573 ÷ x.
Start from the nearest whole number, 24 (24² = 576):
| Step | Guess x | 573 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 23.8750000000 | 23.9375000000 | 4 |
| 2 | 23.9375000000 | 23.9373368146 | 23.9374184073 | 9 |
| 3 | 23.9374184073 | 23.9374184070 | 23.9374184072 | all 10 shown |
The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √573 = 23.9374184072 to every decimal shown.
√573 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √573 the pattern is [23; 1, 14, 1, 46] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √573 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 | 9.4 × 10⁻¹ |
| 24/1 | 24.0000000000 | 6.3 × 10⁻² |
| 359/15 | 23.9333333333 | 4.1 × 10⁻³ |
| 383/16 | 23.9375000000 | 8.2 × 10⁻⁵ |
| 17,977/751 | 23.9374167776 | 1.6 × 10⁻⁶ |
| 18,360/767 | 23.9374185137 | 1.1 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 573y² = 1. Its smallest solution in positive whole numbers is x = 383, y = 16.
√573 in geometry and everyday measurements
- A square garage floor of 573 square feet measures about 23.94 ft (23 ft 11 in) per side, and its corner-to-corner diagonal is √1146 ≈ 33.9 ft.
- 573 is not a sum of two whole-number squares — the prime factor 3 and 191 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √573 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 13 × 20 box, because 2² + 13² + 20² = 573.
Square roots near √573 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √570 | √570 | 23.8747 | No |
| √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 |
- The cube root of 573 is about 8.305865.
- Squaring undoes the root: (√573)² = 573, while 573² = 328,329 — the number whose square root is 573.
Frequently asked questions
What is the square root of 573?
The square root of 573 is √573, about 23.9374184072. The negative root, −23.937418, also squares to 573.
Is the square root of 573 rational or irrational?
Irrational. 573 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 √573 be simplified?
No. 573 = 3 × 191 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √573 rounded to two decimal places?
√573 ≈ 23.94 to two decimal places (23.9 to one, 23.937 to three). Check: 23.94² = 573.1236, close to 573.