Square Root Calculator › Square root of 564
Square Root of 564
√564 ≈ 23.748684
564 is not a perfect square, so √564 is an irrational number that is approximately 23.748684. In simplified radical form, √564 = 2√141.
√564 lies between √529 = 23 and √576 = 24.
| Number | 564 |
| Square root (decimal) | 23.748684 |
| Simplified radical | 2√141 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 564
564 = 2^2 × 3 × 47. This is what determines the simplified radical: for √564 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √564 = 2√141.
Estimating √564 by hand
√564 falls between √529 = 23 and √576 = 24, so it starts at 23.something. The Babylonian method refines that guess by averaging it with 564 ÷ the guess:
| Step | Estimate |
| Start | 23 |
| Step 1 | 23.76087 |
| Step 2 | 23.748687 |
| Step 3 | 23.748684 |
After 3 steps the estimate is 23.748684, matching √564 ≈ 23.748684.
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Frequently asked questions
Is 564 a perfect square?
No, 564 is not a perfect square, so its square root is irrational and approximately 23.748684.
What is the square root of 564 in simplest radical form?
√564 = 2√141 in simplest radical form.