Square Root Calculator › Square root of 565
Square Root of 565
√565 ≈ 23.769729
565 is not a perfect square, so √565 is an irrational number that is approximately 23.769729. In simplified radical form, √565 = √565.
√565 lies between √529 = 23 and √576 = 24.
| Number | 565 |
| Square root (decimal) | 23.769729 |
| Simplified radical | √565 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 565
565 = 5 × 113. This is what determines the simplified radical: for √565 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √565 is already in simplest radical form as √565.
Estimating √565 by hand
√565 falls between √529 = 23 and √576 = 24, so it starts at 23.something. The Babylonian method refines that guess by averaging it with 565 ÷ the guess:
| Step | Estimate |
| Start | 23 |
| Step 1 | 23.782609 |
| Step 2 | 23.769732 |
| Step 3 | 23.769729 |
After 3 steps the estimate is 23.769729, matching √565 ≈ 23.769729.
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Frequently asked questions
Is 565 a perfect square?
No, 565 is not a perfect square, so its square root is irrational and approximately 23.769729.
What is the square root of 565 in simplest radical form?
√565 = √565 in simplest radical form.