√576 at a glance
- Exact value
- 24
- Decimal (10 places)
- 24
- Rounded
- 24
- Perfect square?
- Yes — 24²
- Rational?
- Rational (whole number)
- Both square roots
- ±24
- Prime factorization
- 2⁶ × 3²
- Cube root
- 8.320335
How to simplify √576
Write 576 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√576 = 2³ × 3 = 24
576 has 11 factor pairs (1 × 576, 2 × 288, 3 × 192, 4 × 144, 6 × 96, 8 × 72, 9 × 64, 12 × 48, 16 × 36, 18 × 32, 24 × 24). The pair with two equal numbers, 24 × 24, is what makes it a perfect square — and why it has an odd number of factors.
Every perfect square is also a sum of consecutive odd numbers starting at 1. Here 576 = 1 + 3 + 5 + 7 + … + 47, the first 24 odd numbers.
Where √576 sits between perfect squares
576 is 24². The perfect squares on either side are 529 (23²) and 625 (25²), so the gaps are 47 below and 49 above — consecutive squares always differ by consecutive odd numbers.
Finding √576 with the Babylonian method
Picture a rectangle with an area of 576 and one side x; the other side must be 576 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √576.
To show the method at work on a perfect square, start one above the answer, at 25:
| Step | Guess x | 576 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 23.0400000000 | 24.0200000000 | 1 |
| 2 | 24.0200000000 | 23.9800166528 | 24.0000083264 | 5 |
| 3 | 24.0000083264 | 23.9999916736 | 24.0000000000 | all 10 shown |
The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √576 = 24.0000000000 to every decimal shown.
√576 in geometry and everyday measurements
- A square garage floor of 576 square feet measures exactly 24 ft (24 ft) per side, and its corner-to-corner diagonal is √1152 ≈ 33.9 ft.
- 576 is not a sum of two whole-number squares — no pair of squares adds up to it — so √576 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 8 × 16 × 16 box, because 8² + 16² + 16² = 576.
Square roots near √576 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √578 | 17√2 | 24.0416 | No |
| √579 | √579 | 24.0624 | No |
- The cube root of 576 is about 8.320335.
- Taking the square root twice gives the fourth root: √24 ≈ 4.898979.
- Because 576 = 4 × 144, the root is twice √144: 2 × 12 ≈ 24.
Frequently asked questions
What is the square root of 576?
The square root of 576 is 24, because 24 × 24 = 576. Strictly, 576 has two square roots, 24 and −24; the √ symbol means the positive one.
Is the square root of 576 rational or irrational?
Rational. 576 is a perfect square, so its square root is the whole number 24.
Can √576 be simplified?
Yes, all the way to a whole number: √576 = 24.
What is √576 rounded to two decimal places?
√576 is exactly 24, or 24.00 written to two decimal places.