Square Root Calculator › Square root of 517
Square Root of 517
√517 ≈ 22.737634
517 is not a perfect square, so √517 is an irrational number that is approximately 22.737634. In simplified radical form, √517 = √517.
√517 lies between √484 = 22 and √529 = 23.
| Number | 517 |
| Square root (decimal) | 22.737634 |
| Simplified radical | √517 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 517
517 = 11 × 47. This is what determines the simplified radical: for √517 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √517 is already in simplest radical form as √517.
Estimating √517 by hand
√517 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 517 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.75 |
| Step 2 | 22.737637 |
| Step 3 | 22.737634 |
After 3 steps the estimate is 22.737634, matching √517 ≈ 22.737634.
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Frequently asked questions
Is 517 a perfect square?
No, 517 is not a perfect square, so its square root is irrational and approximately 22.737634.
What is the square root of 517 in simplest radical form?
√517 = √517 in simplest radical form.